Operations on Real Numbers & Laws of Exponents

Mixing numbers is like mixing playlists — sometimes you create something entirely new, and sometimes you land right back on a familiar track. When you add and multiply surds (roots like √2, √3), the results follow neat but surprising rules, including the famous moment when two irrationals multiply to give a plain rational. This lesson lays out those rules, the surd laws you must memorise, and the errors that cost easy marks.

Definition: A surd is the irrational root of a rational number, such as √2, √3, ∛5. (Roots that come out exact, like √9 = 3, are not surds.)

Definition: A rational number is one expressible as p/q (integers, q ≠ 0); an irrational number is one that is not.

The three combination rules

When you combine a rational with an irrational, the outcome is predictable:

• rational + irrational = irrational. For example 2 + √3 is irrational. Adding a "tidy" number cannot smooth out the endless non-repeating tail of √3.
• rational − irrational = irrational (same reasoning), e.g. 5 − √2 is irrational.
• (non-zero) rational × irrational = irrational. For example 5√2 is irrational. Scaling an irrational by an ordinary number keeps it irrational — unless you multiply by 0, which gives 0 (rational).

Why it matters: These rules let you instantly classify expressions like 3 + 2√5 or 7√2 as irrational without computing decimals — a common exam shortcut. You can even prove them by contradiction: if 2 + √3 were rational, then √3 = (that rational) − 2 would be rational too, which is false.

The surd surprise: irrational × irrational can be rational

Here is the twist that catches students out. Multiplying two irrationals does not always give an irrational:

√2 × √2 = √4 = 2 — a rational number!

Two irrationals collapsed into a plain integer. This happens whenever the roots "complete" a perfect square. It also means irrational × irrational is unpredictable: √2 × √3 = √6 (irrational), but √2 × √8 = √16 = 4 (rational). You must actually compute it; there is no blanket rule.

The surd laws you must know

For non-negative a and b:

• √a × √b = √(ab) — multiply under one root. So √2 × √3 = √6.
• √a ÷ √b = √(a/b) — divide under one root (b ≠ 0). So √6 ÷ √2 = √3.
• (√a)² = a — squaring undoes the root. So (√5)² = 5.
• √(a²b) = a√b — pull out perfect-square factors. So √50 = √(25·2) = 5√2.

Question: Simplify √2 × √3 × √6.
Solution:
Step 1: Combine the first two: √2 × √3 = √(2×3) = √6.
Step 2: Multiply by the last: √6 × √6 = √(6×6) = √36.
Step 3: Evaluate: √36 = 6.
Conclusion: √2 × √3 × √6 = 6, a rational number.

Question: Simplify √48 + √12.
Solution:
Step 1: Pull out perfect squares: √48 = √(16·3) = 4√3.
Step 2: Likewise √12 = √(4·3) = 2√3.
Step 3: Now both have the same surd √3, so add the coefficients: 4√3 + 2√3 = 6√3.
Conclusion: √48 + √12 = 6√3. (You may only add surds directly when they share the same root.)

Real-world example: Suppose two square plots have areas 2 m² and 8 m². Their sides are √2 m and √8 m = 2√2 m. The combined side-length if laid end to end is √2 + 2√2 = 3√2 m — you add the coefficients of the common surd, not the numbers under the root. Geometry keeps you honest: you cannot just shove everything under one root sign.

Common misconception: "√a + √b = √(a + b)." Completely false. Check it: √4 + √9 = 2 + 3 = 5, but √(4 + 9) = √13 ≈ 3.61. Roots multiply and divide inside a single radical, but they do not add or subtract inside one.

Common misconception: "Multiplying two irrationals always gives an irrational." No — √2 × √2 = 2 is rational. The product depends on the specific numbers.

Common misconception: "You can add any two surds directly." Only like surds (same number under the root) combine: 3√5 + 2√5 = 5√5, but 3√5 + 2√2 cannot be simplified into a single surd — leave it as is.

Having a square root sitting in the denominator of a fraction is perfectly correct mathematically, yet mathematicians almost always rewrite it so the bottom is a clean whole number. This lesson explains what rationalising the denominator means, exactly why we do it, the two standard techniques (multiplying by a surd and multiplying by a conjugate), and how to handle every case you will meet in Class 9 number systems and beyond.

Definition: A surd is an irrational root that cannot be simplified to a rational number, such as √2, √3 or ∛5. A number like √4 = 2 is not a surd because it simplifies to a rational.

Definition: Rationalising the denominator means rewriting a fraction in an equal form whose denominator contains no surd (no root sign), so the bottom is a rational number.

Why we bother rationalising

Mathematically, 1/√2 and √2/2 are exactly the same number (both equal about 0.7071). So why convert? There are real reasons.

Before calculators were common, people divided by hand. Dividing 1 by 1.41421356… is painful and error-prone. But √2/2 just means "take √2 ≈ 1.41421356 and halve it" — dividing by the integer 2 is trivial. A rational denominator turns a hard division into an easy one.

There is also a deeper reason: a standard form. Mathematics likes every answer written one agreed way so two students who solve the same problem get visibly identical answers. The convention is "no surds downstairs." When your textbook answer key gives √2/2 but you wrote 1/√2, both are right, but you only earn full marks by presenting the standard rationalised form.

Why it matters: In later algebra, calculus and physics you constantly add and compare fractions. A common rational denominator lets you combine fractions cleanly; a messy surd denominator blocks that.

The core identity that powers everything

Two algebraic identities do all the heavy lifting:

• (√a)(√a) = a — a square root times itself removes the root.
• (a + b)(a − b) = a² − b² — the difference-of-squares identity.

The second identity is the magic wand for two-term denominators. When you multiply a sum of two terms by the same expression with the middle sign flipped, all the cross terms cancel and you are left only with squares — and squaring a square root destroys it.

Case 1 — a single surd in the denominator

When the denominator is a single surd, multiply the top and bottom by that surd. Because you multiply by (surd/surd), which equals 1, the value of the fraction is unchanged — only its appearance changes.

Question: Rationalise 1/√2.
Solution:
Step 1: Multiply numerator and denominator by √2: (1/√2) × (√2/√2).
Step 2: Numerator becomes √2; denominator becomes √2 × √2 = 2.
Conclusion: 1/√2 = √2/2. The root is now in the numerator and the denominator is the rational number 2.

Question: Rationalise 5/(2√3).
Solution:
Step 1: The surd part is √3, so multiply top and bottom by √3: (5/2√3) × (√3/√3).
Step 2: Numerator = 5√3; denominator = 2 × √3 × √3 = 2 × 3 = 6.
Conclusion: 5/(2√3) = 5√3/6.

Case 2 — two terms (using the conjugate)

When the denominator has two terms, like (√3 + 1) or (5 − √2), you cannot just multiply by one surd — you would create new cross terms. Instead multiply by the conjugate.

Definition: The conjugate of a two-term expression is the same expression with the sign between the terms reversed. The conjugate of (√3 + 1) is (√3 − 1); the conjugate of (5 − √2) is (5 + √2).

Multiplying an expression by its conjugate triggers the difference-of-squares identity, squaring each term and wiping out the roots.

Question: Rationalise 1/(√3 + 1).
Solution:
Step 1: Multiply numerator and denominator by the conjugate (√3 − 1).
Step 2: Denominator = (√3 + 1)(√3 − 1) = (√3)² − 1² = 3 − 1 = 2.
Step 3: Numerator = 1 × (√3 − 1) = √3 − 1.
Conclusion: 1/(√3 + 1) = (√3 − 1)/2.

Question: Rationalise 1/(√5 − √2).
Solution:
Step 1: Conjugate is (√5 + √2). Multiply top and bottom by it.
Step 2: Denominator = (√5 − √2)(√5 + √2) = (√5)² − (√2)² = 5 − 2 = 3.
Step 3: Numerator = √5 + √2.
Conclusion: 1/(√5 − √2) = (√5 + √2)/3.

A worked example combining ideas

Question: Rationalise and simplify (3 + √2)/(3 − √2).
Solution:
Step 1: Multiply numerator and denominator by the conjugate (3 + √2).
Step 2: Denominator = (3 − √2)(3 + √2) = 3² − (√2)² = 9 − 2 = 7.
Step 3: Numerator = (3 + √2)(3 + √2) = 9 + 6√2 + 2 = 11 + 6√2.
Conclusion: (3 + √2)/(3 − √2) = (11 + 6√2)/7.

Common misconception: "Multiply by the same thing top and bottom — won't that change the answer?" No. You are multiplying by a form of 1 (surd/surd or conjugate/conjugate), and multiplying by 1 never changes a number's value, only its written form.

Common misconception: "The conjugate of (√3 + 1) is (1 + √3)." Reordering the terms does nothing — both still have a plus sign, so the roots will not cancel. The conjugate must flip the sign: (√3 − 1).

Common misconception: "Rationalising makes the number rational." It does not. (√3 − 1)/2 is still irrational. Only the denominator becomes rational; the overall number keeps its irrational character.

In real life, engineers and physicists rationalise constants so repeated calculations divide by clean integers, reducing rounding errors when the same expression is evaluated thousands of times in a simulation.

Exponents are a compact language for repeated multiplication, and once you know their handful of rules you can simplify expressions that would otherwise take pages. This lesson covers the laws of exponents for real numbers, why each law is true (not just what it says), how exponents extend to zero, negative and fractional powers, and several worked examples that show the rules working together.

Definition: In the expression aᵐ, the number a is the base and m is the exponent (or power or index). It means a multiplied by itself m times: a³ = a × a × a.

Definition: The laws of exponents are a set of rules that tell you how to combine powers when you multiply, divide or raise them — valid here for any positive real base a and rational exponents m and n.

The five core laws and the intuition behind them

The rules are not arbitrary; each one drops straight out of the meaning of "repeated multiplication."

Product law: aᵐ · aⁿ = a^(m+n). Multiplying two powers of the same base means writing the base out m times and then n more times — m + n copies in total. For example a² · a³ = (a·a)(a·a·a) = a⁵, and indeed 2 + 3 = 5. You add the exponents.

Quotient law: aᵐ ÷ aⁿ = a^(m−n). Division cancels copies. a⁵ ÷ a² = (a·a·a·a·a)/(a·a) — two factors cancel, leaving a³, and 5 − 2 = 3. You subtract the exponents.

Power of a power: (aᵐ)ⁿ = a^(mn). Here aᵐ is raised to the nth power, i.e. multiplied by itself n times. Each copy contributes m factors, so you get m × n factors total. (a²)³ = a² · a² · a² = a⁶, and 2 × 3 = 6. You multiply the exponents.

Power of a product: aᵐ · bᵐ = (ab)ᵐ. When two different bases share the same exponent, you may group them. 2³ · 5³ = (2·5)³ = 10³ = 1000. This is just rearranging the order of multiplication.

Power of a quotient: aᵐ ÷ bᵐ = (a/b)ᵐ. The same grouping idea applied to division.

Why it matters: These five laws let you collapse huge products and divisions into a single power, which is the backbone of scientific notation, logarithms, exponential growth and computer science (binary sizes).

Zero and negative exponents

Zero exponent: a⁰ = 1 (for any a ≠ 0). Why? Use the quotient law: aⁿ ÷ aⁿ = a^(n−n) = a⁰. But any nonzero number divided by itself is 1. So a⁰ must equal 1 for the laws to stay consistent. (0⁰ is left undefined.)

Negative exponent: a^(−n) = 1/aⁿ. Again from the quotient law: a⁰ ÷ aⁿ = a^(0−n) = a^(−n), and a⁰ ÷ aⁿ = 1/aⁿ. So a negative exponent simply means "reciprocal." For example 2^(−3) = 1/2³ = 1/8.

Why it matters: Negative powers are how very small numbers are written compactly — the mass of an electron is about 9.1 × 10⁻³¹ kg.

Fractional exponents are roots

This is the bridge from exponents to surds. We define a^(1/n) = ⁿ√a, the nth root of a. The definition is forced by the power-of-a-power law: (a^(1/n))ⁿ = a^(n/n) = a¹ = a, so a^(1/n) is exactly the number whose nth power is a — that is the definition of the nth root.

More generally, a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ). For example 8^(2/3) = (∛8)² = 2² = 4.

Real-world example: Computer storage runs on powers of two. 2¹⁰ = 1024, which is why one kilobyte is 1024 bytes, a megabyte is 2²⁰ bytes, and a gigabyte is 2³⁰ bytes. Compound interest, population growth and radioactive decay all run on the same exponent rules.

Worked examples

Question: Simplify 7² · 7³ ÷ 7⁴.
Solution:
Step 1: Apply the product law on top: 7² · 7³ = 7^(2+3) = 7⁵.
Step 2: Apply the quotient law: 7⁵ ÷ 7⁴ = 7^(5−4) = 7¹.
Conclusion: The expression equals 7.

Question: Evaluate 16^(3/4).
Solution:
Step 1: 16^(3/4) = (⁴√16)³.
Step 2: ⁴√16 = 2 (since 2⁴ = 16).
Step 3: 2³ = 8.
Conclusion: 16^(3/4) = 8.

Question: Simplify (3² · 3⁵)^(1/7).
Solution:
Step 1: Inside the bracket, product law: 3² · 3⁵ = 3⁷.
Step 2: Power of a power: (3⁷)^(1/7) = 3^(7 × 1/7) = 3¹.
Conclusion: The expression equals 3.

Common misconception: "aᵐ · aⁿ = a^(m·n)." Wrong — when multiplying powers you add exponents; you only multiply exponents in (aᵐ)ⁿ. So 2² · 2³ = 2⁵ = 32, not 2⁶.

Common misconception: "a negative exponent makes the number negative." No. 2^(−3) = 1/8, a positive number. The minus sign signals a reciprocal, not a sign change.

Common misconception: "The laws need the same base AND same exponent." The product and quotient laws need the same base; the power-of-a-product law needs the same exponent. Mixing these up causes most exam errors.

Once you accept that irrational numbers like √2 and √3 are genuine numbers, the natural next question is: how do you do arithmetic with them, and what rules govern powers and roots? This lesson covers operations on real numbers, the surd identities, the all-important trick of rationalising a denominator, and the laws of exponents extended to rational powers.

Definition: A surd is an irrational root such as √2, √3, or ∛5 — a root that cannot be simplified to a rational number.

Real numbers are closed under arithmetic

You can add, subtract, multiply, and divide (except by 0) any two real numbers and always land back on a real number. Within this, the rationals form a tidy inner world: a rational ± a rational is rational, and a rational × a rational is rational.

The interesting cases involve irrationals:

• Rational + irrational = irrational. For example, 2 + √3 is irrational. (If it were rational, then subtracting the rational 2 would make √3 rational — a contradiction.)
• Non-zero rational × irrational = irrational. For example, 5√2 is irrational.

Why it matters: These facts let you instantly classify expressions as irrational without computing decimals — a frequent exam shortcut.

Two irrationals are unpredictable

Combining two irrationals can give either a rational or an irrational — it depends on the specific numbers.

• √2 × √2 = 2 (rational).
• √2 × √3 = √6 (irrational).
• (3 + √2) + (3 − √2) = 6 (rational), while (3 + √2) + √5 stays irrational.

Common misconception: "Any two irrationals multiply to an irrational." Wrong — √2 × √8 = √16 = 4 is rational. You must actually combine them and check.

Surd identities for square roots

For non-negative reals a and b:

• √(ab) = √a × √b
• √(a/b) = √a / √b (b ≠ 0)
• (√a + √b)(√a − √b) = a − b
• (a + √b)(a − √b) = a² − b

That third identity is the difference-of-squares pattern, and it is the heart of rationalisation, because it deletes the square root.

Common misconception: "√2 + √3 = √5." This is WRONG — you cannot add quantities under separate root signs. Numerically √2 + √3 ≈ 1.414 + 1.732 = 3.146, while √5 ≈ 2.236. Roots add only when the radicand is identical (e.g. 2√3 + 5√3 = 7√3).

Rationalising a denominator

Definition: Rationalising means rewriting a fraction so that no surd remains in the denominator, by multiplying top and bottom by a suitable conjugate (the same two terms with the opposite sign).

Why it matters: A rational denominator makes numbers far easier to estimate, add, and compare — historically essential before calculators, and still the standard "simplest form" examiners expect.

Question: Rationalise 1/(2 + √3).
Solution:
Step 1: The conjugate of (2 + √3) is (2 − √3). Multiply numerator and denominator by it.
Step 2: Denominator = (2 + √3)(2 − √3) = 2² − (√3)² = 4 − 3 = 1.
Step 3: So 1/(2 + √3) = (2 − √3)/1.
Conclusion: 1/(2 + √3) = 2 − √3.

Question: Rationalise 5/(√3 − √2).
Solution:
Step 1: Conjugate is (√3 + √2); multiply top and bottom by it.
Step 2: Denominator = (√3)² − (√2)² = 3 − 2 = 1.
Step 3: Numerator = 5(√3 + √2).
Conclusion: 5/(√3 − √2) = 5(√3 + √2) = 5√3 + 5√2.

Laws of exponents (now with rational powers)

For a > 0, b > 0 and rational exponents p, q:

• aᵖ × aᑫ = a^(p+q)
• aᵖ / aᑫ = a^(p−q)
• (aᵖ)ᑫ = a^(pq)
• aᵖ × bᵖ = (ab)ᵖ
• a⁰ = 1 and a^(−p) = 1/aᵖ

Fractional powers are simply roots: a^(1/n) = the nth root of a, and a^(m/n) = the nth root of (aᵐ) = (nth root of a)ᵐ.

Real-world example: Compound interest, population growth, and sound-intensity (decibel) calculations all rely on these exponent laws to combine repeated multiplications cleanly.

Question: Evaluate 8^(2/3).
Solution:
Step 1: Write 8 = 2³.
Step 2: 8^(2/3) = (2³)^(2/3) = 2^(3 × 2/3) = 2².
Conclusion: 8^(2/3) = 4.

Common misconception: "a^(m/n) means a^m ÷ n." Wrong — the n is a root, not a divisor. a^(3/2) = √(a³), not a³ ÷ 2.

Fractions with a surd in the denominator look messy and are hard to estimate — so we "clean" them by rationalising. This lesson walks through one classic worked example, 1/(√7 − √3), in full detail, and shows you exactly why the conjugate trick works every time.

Definition: To rationalise a denominator is to rewrite a fraction so that no irrational root remains below the line, by multiplying the top and bottom by the conjugate of the denominator.

Why the conjugate works

The denominator √7 − √3 is a difference of two square roots. Its conjugate is √7 + √3 — same terms, opposite sign. When you multiply a difference by its matching sum you get the difference of squares:
(√a − √b)(√a + √b) = (√a)² − (√b)² = a − b.

The square roots vanish because squaring a square root removes it. That is the whole secret: the conjugate converts an irrational denominator into a plain rational number. Multiplying top and bottom by the same quantity does not change the value of the fraction (you are multiplying by a disguised 1).

Worked example

Question: Rationalise the denominator of 1/(√7 − √3).
Solution:
Step 1: The conjugate of (√7 − √3) is (√7 + √3). Multiply numerator and denominator by it:
1/(√7 − √3) = [1 × (√7 + √3)] / [(√7 − √3)(√7 + √3)].
Step 2: Expand the denominator using (a − b)(a + b) = a² − b²:
(√7)² − (√3)² = 7 − 3 = 4.
Step 3: The expression becomes (√7 + √3)/4.
Conclusion: 1/(√7 − √3) = (√7 + √3)/4.

Checking the answer numerically

Why it matters: A quick decimal check catches sign and arithmetic slips before you lose marks.

√7 ≈ 2.6458 and √3 ≈ 1.7321, so √7 − √3 ≈ 0.9137, giving 1/0.9137 ≈ 1.0944.
The answer (2.6458 + 1.7321)/4 = 4.3779/4 ≈ 1.0945. The two match, confirming the result.

Common misconceptions

Common misconception: "Multiply by (√7 − √3) again." Wrong — multiplying by the same expression gives (√7 − √3)², which still contains the cross-term −2√21, so a surd remains. You must use the conjugate (opposite sign) so the cross-terms cancel.

Common misconception: "(√7 − √3)(√7 + √3) = 7 − 3√21 + ... ." Wrong — the middle terms (−√21 and +√21) cancel exactly, leaving only 7 − 3 = 4. That clean cancellation is precisely why the method works.

Expressions like 16^(3/4) × 8^(−2/3) look intimidating until you spot the trick: rewrite every base as a power of the same prime, and the laws of exponents do the rest. This lesson works that example end to end and explains the reasoning behind each step.

Definition: A rational exponent a^(m/n) means the nth root of a raised to the m: a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ. A negative exponent means a reciprocal: a^(−p) = 1/aᵖ.

The key idea: share a common base

16 and 8 are both powers of 2 (16 = 2⁴, 8 = 2³). Once both factors are written with base 2, you can use a^(m)^n = a^(mn) to flatten the powers, then a^p × a^q = a^(p+q) to combine. Choosing a common base turns a hard-looking product into simple integer arithmetic on exponents.

Why it matters: Almost every "simplify the exponent" problem in exams hides a common base (2, 3, 5, or 10). Spotting it is the single most useful move in this topic.

Worked example

Question: Simplify 16^(3/4) × 8^(−2/3).
Solution:
Step 1: Write the bases as powers of 2: 16 = 2⁴ and 8 = 2³.
Step 2: Apply (aᵐ)ⁿ = a^(mn) to each factor.
16^(3/4) = (2⁴)^(3/4) = 2^(4 × 3/4) = 2³.
8^(−2/3) = (2³)^(−2/3) = 2^(3 × (−2/3)) = 2^(−2).
Step 3: Multiply using aᵖ × aᑫ = a^(p+q):
2³ × 2^(−2) = 2^(3 + (−2)) = 2¹ = 2.
Conclusion: 16^(3/4) × 8^(−2/3) = 2.

Verifying with roots directly

Why it matters: Computing each piece as an actual root cross-checks the exponent algebra.

16^(3/4) = (⁴√16)³ = 2³ = 8.
8^(2/3) = (∛8)² = 2² = 4, so 8^(−2/3) = 1/4.
Then 8 × (1/4) = 2. Confirmed.

Common misconceptions

Common misconception: "(2⁴)^(3/4) = 2^(4 + 3/4)." Wrong — a power of a power multiplies the exponents, it does not add them: (aᵐ)ⁿ = a^(mn), so (2⁴)^(3/4) = 2^(4 × 3/4) = 2³.

Common misconception: "A negative exponent makes the number negative." Wrong — it makes a reciprocal. 8^(−2/3) = 1/4, which is positive, not −4 or −1/4-with-a-sign-issue. The base 8 stays positive throughout.

Almost every algebra problem you will meet in Class 9 and beyond eventually collapses into one of two skills: cleaning up a messy root (a surd) or shuffling powers around (an exponent). Master the handful of identities below and you can simplify, rationalise, and compute with confidence.

Definition: A surd is an irrational number written using a root sign, such as root 2, root 3, or root 7, whose exact decimal value never terminates or repeats.
Definition: An exponent (or index) tells you how many times a number, called the base, is multiplied by itself; in a^p, a is the base and p is the exponent.

The surd identities and why they hold

For all a, b that are greater than or equal to 0 (and denominators non-zero):

• root(ab) = root a times root b
• root(a/b) = root a / root b
• (root a + root b)(root a - root b) = a - b
• (a + root b)(a - root b) = a^2 - b

The first identity is not a definition pulled from thin air; it follows from the laws of exponents. Since root x means x^(1/2), we have (ab)^(1/2) = a^(1/2) times b^(1/2), which is exactly root a times root b. The same logic gives the quotient rule. These two let you split a root apart to simplify it: root 50 = root(25 times 2) = root 25 times root 2 = 5 root 2. Pulling out the largest perfect-square factor (here 25) is the standard simplification move.

The last two identities are just the algebraic identity (x + y)(x - y) = x^2 - y^2 in disguise. In (root a + root b)(root a - root b), put x = root a and y = root b; then x^2 = a and y^2 = b, so the product is a - b. The two surds vanish, leaving a clean rational number. This is the engine behind rationalisation.

Why it matters: Splitting roots simplifies; combining roots via the difference-of-squares pattern destroys roots. Knowing which identity does which is the whole game.

Rationalising factors (conjugates)

Definition: The rationalising factor (or conjugate) of a surd expression is the partner you multiply by to wipe the root out of a denominator.

• For (a + root b) use (a - root b)
• For (root a + root b) use (root a - root b)

Why it matters: A number like 1/(root 2) is awkward to estimate and to add to other fractions. Convention demands a rational denominator. You multiply top and bottom by the conjugate, which changes the form but not the value (because you are multiplying by 1).

Question: Rationalise the denominator of 1/(3 + root 5).
Solution:
Step 1: Multiply numerator and denominator by the conjugate (3 - root 5).
Step 2: Denominator becomes (3 + root 5)(3 - root 5) = 3^2 - 5 = 9 - 5 = 4.
Step 3: Numerator becomes 1 times (3 - root 5) = 3 - root 5.
Conclusion: 1/(3 + root 5) = (3 - root 5)/4. The denominator is now rational.

Laws of exponents

For a > 0, b > 0 and rational p, q:

• a^p times a^q = a^(p+q)
• a^p / a^q = a^(p-q)
• (a^p)^q = a^(pq)
• a^p times b^p = (a b)^p
• a^0 = 1, and a^(-p) = 1 / a^p

The product rule is just counting: a^2 times a^3 means (a times a)(a times a times a), which is five a's multiplied, i.e. a^5. The quotient rule is the same count run backwards, cancelling common factors. The power rule (a^p)^q raises an already-formed power again, multiplying the exponents. The rule a^0 = 1 is forced on us by consistency: a^p / a^p must equal 1, but by the quotient rule it equals a^(p-p) = a^0, so a^0 = 1. Negative exponents simply mean reciprocals.

Fractional (rational) exponents

This is where surds and exponents merge into one language:

• a^(1/n) = nth root of a
• a^(m/n) = nth root of (a^m) = (nth root of a)^m

The definition is chosen so the power rule keeps working: (a^(1/n))^n = a^(n times 1/n) = a^1 = a, and the only number whose nth power is a is the nth root of a. Once you accept a^(1/n) as a root, a^(m/n) is just that root raised to the m, and the two orders (root-then-power or power-then-root) always agree.

Question: Evaluate 8^(2/3).
Solution:
Step 1: 8^(2/3) = (cube root of 8)^2.
Step 2: cube root of 8 = 2.
Step 3: 2^2 = 4.
Conclusion: 8^(2/3) = 4.

Real-world example: Compound growth and "doubling time" use fractional exponents. If money grows by a factor of 4 over 3 years, the per-year growth factor is 4^(1/3), roughly 1.587 — i.e. about 58.7% a year — which you compute exactly the way you evaluate 8^(2/3).

Common misconception: Students write root a + root b = root(a+b). This is false. root 9 + root 16 = 3 + 4 = 7, but root(9+16) = root 25 = 5. The split rule works only across multiplication and division, never across addition or subtraction.

When you add, subtract, multiply, or divide real numbers, the answer is sometimes guaranteed to be rational, sometimes guaranteed to be irrational, and sometimes it genuinely depends on the numbers. Knowing which case you are in saves you from both careless errors and unnecessary work.

Definition: A rational number can be written as p/q where p and q are integers and q is not zero; its decimal either terminates or repeats.
Definition: An irrational number cannot be written as such a fraction; its decimal goes on forever without repeating (e.g. root 2, root 3, pi).

How rationals and irrationals combine

The most important behaviour rules, with the reasoning behind each:

1. rational +/- rational and rational x rational always stay rational. This is because fractions are closed under these operations: p/q + r/s = (ps + rq)/(qs), still a ratio of integers.
2. rational + irrational is irrational, and a non-zero rational x irrational is irrational. The proof is by contradiction: if 3 + root 2 were rational, then root 2 = (that rational) - 3 would also be rational, which is false. The zero multiplier is the lone exception, since 0 times anything is 0 (rational).
3. Two irrationals can combine to either a rational or an irrational — it depends on the numbers. root 2 times root 2 = 2 (rational), but root 2 times root 3 = root 6 (irrational). Likewise (2 + root 2) + (2 - root 2) = 4 (rational), while root 2 + root 3 stays irrational.

Why it matters: In exams you are often asked "is this number rational or irrational?" without computing it fully. Rule 2 instantly tells you 5 + root 7 is irrational; rule 3 warns you that you must actually look at the numbers before deciding about a product of two surds.

Real-world example: Measurement. A square of side 1 metre has diagonal root 2 metres — irrational. Two such diagonals laid end to end give 2 root 2 metres (still irrational, rule 2), but the area of the square is root 2 times root 2 = 2 square metres (rational, rule 3). The geometry forces specific surds to multiply back to neat rationals.

Surd rules for simplification

For a, b greater than or equal to 0:

• root(ab) = root a x root b
• root(a/b) = root a / root b
• (root a + root b)(root a - root b) = a - b

The first two let you simplify by pulling perfect-square factors out: root 18 = root(9 times 2) = 3 root 2. The third is the difference-of-squares identity and is the basis of rationalising denominators.

Rationalising the denominator

Definition: To rationalise a denominator is to rewrite a fraction so that no surd remains downstairs, by multiplying top and bottom by a suitable factor.

For 1/root a, multiply by root a / root a. For 1/(a + root b), multiply by the conjugate (a - root b).

Question: Rationalise 5/(root 7 - root 2).
Solution:
Step 1: Multiply top and bottom by the conjugate (root 7 + root 2).
Step 2: Denominator = (root 7 - root 2)(root 7 + root 2) = 7 - 2 = 5.
Step 3: Numerator = 5(root 7 + root 2).
Conclusion: 5/(root 7 - root 2) = 5(root 7 + root 2)/5 = root 7 + root 2.

Laws of exponents and rational exponents

For a > 0 and rational p, q:

• a^p x a^q = a^(p+q)
• a^p / a^q = a^(p-q)
• (a^p)^q = a^(pq)
• a^p x b^p = (ab)^p

Rational exponents are simply roots:

• a^(1/n) = nth root of a
• a^(m/n) = nth root of a^m

This unifies surds and powers: root 2 is just 2^(1/2), so every surd rule is really an exponent rule in disguise.

Question: Simplify 7^(1/2) x 7^(3/2).
Solution:
Step 1: Same base, so add exponents: 7^(1/2 + 3/2) = 7^(4/2).
Step 2: 7^2 = 49.
Conclusion: The product is 49 — a rational result, illustrating rule 3 (two irrational factors giving a rational).

Common misconception: Believing root 2 + root 3 = root 5. It does not. You may not add quantities sitting under separate root signs. Numerically root 2 + root 3 is about 1.414 + 1.732 = 3.146, whereas root 5 is about 2.236. Roots combine only over multiplication and division.

Cover the answer, recall, then check. 8 cards on operations and exponent laws.

Q1. What does it mean to rationalise a denominator?
A1. To remove the surd (irrational number) from the denominator by multiplying by a suitable factor.

Q2. Rationalise 1/√2.
A2. Multiply top and bottom by √2: 1/√2 = √2/2.

Q3. What is the rationalising factor of (√3 + 2)?
A3. Its conjugate, (√3 − 2).

Q4. State the product law of exponents: aᵐ × aⁿ = ?
A4. a^(m+n).

Q5. Simplify: aᵐ ÷ aⁿ.
A5. a^(m−n).

Q6. What is the value of a⁰ (a ≠ 0)?
A6. 1.

Q7. Evaluate 2^(1/2) × 2^(1/2).
A7. 2^(1/2 + 1/2) = 2¹ = 2.

Q8. Write √a in exponent (power) form.
A8. a^(1/2).