Learn · Mathematics · Class 9

Real Numbers & Indices

The number system, surds and index laws, the foundation everything else rests on.

11 parts

Scroll to begin ↓

📘 DefinitionThe real number systemPart 1 of 11

🔍 In detailProving √2 is irrationalPart 2 of 11

🧩 The partsThe index lawsPart 3 of 11

✏️ Worked exampleWorking with fractional indicesPart 4 of 11

🔍 In detailSurdsPart 5 of 11

✏️ Worked exampleRationalising the denominatorPart 6 of 11

✏️ Worked exampleStandard form in calculationsPart 7 of 11

🌍 In the real worldWhy this foundation mattersPart 8 of 11

⚠️ Common mistakesWhere marks are lostPart 9 of 11

✅ SummaryWhat to rememberPart 10 of 11

✏️ Worked exampleWatch one worked throughPart 11 of 11

1📘 DefinitionPart 1 / 11

The real number system

The real numbers are all rationals and irrationals together, every point on the number line.

Rational: writable as p/q with q ≠ 0. Decimals that terminate or recur.
Irrational: cannot be written as a fraction. Decimals that never end and never repeat.

Each set nests inside the next: naturals ⊂ integers ⊂ rationals ⊂ reals.

√2, √3, π and e are irrational; √9 is not, because it equals 3.

2🔍 In detailPart 2 / 11

Proving √2 is irrational

A standard proof and a good example of proof by contradiction.

Assume √2 = p/q in lowest terms, so p and q share no common factor.

Squaring: 2 = p²/q², so p² = 2q². Therefore p² is even, so p is even: because an odd number squared is always odd.

Write p = 2k. Then 4k² = 2q², so q² = 2k², meaning q is even too.

Contradiction: we assumed no common factor, yet both are even.

The assumption must be false, so √2 cannot be a fraction. ∎

Why this is worth knowing: it shows a claim about all possible fractions can be settled without checking any of them.

3🧩 The partsPart 3 / 11

The index laws

a^m × aⁿ = a^m+n
a^m ÷ aⁿ = a^m−n
(a^m)ⁿ = a^mn
(ab)ⁿ = aⁿbⁿ
a⁰ = 1
a⁻ⁿ = 1/aⁿ
a^1/n = ⁿ√a a^m/n = (ⁿ√a)^m

Same base only. And a negative index never makes the answer negative.

4✏️ Worked examplePart 4 / 11

Working with fractional indices

Question. Evaluate 27^−2/3.

Take it one piece at a time.

Step 1: the negative means reciprocal.

27^−2/3 = 1 / 27^2/3

Step 2: the denominator of the fraction is the root. Take it first.

∛27 = 3

Step 3: the numerator is the power.

3² = 9

Step 4: combine.

= 1/9

Why take the root first: cubing 27 gives 19,683 before you find its cube root. Same answer, far larger numbers, far more chance of error.

5🔍 In detailPart 5 / 11

Surds

A surd is an irrational root left in exact form.

Simplifying: extract any square factor.

√72 = √(36 × 2) = 6√2

Multiplying and dividing:

√a × √b = √(ab)
√a ÷ √b = √(a/b)

Adding, only like surds combine, exactly like algebraic terms.

3√5 + 2√5 = 5√5 ✅
√2 + √3, cannot be combined

The classic error: √a + √b ≠ √(a+b). Check: √9 + √16 = 7, but √25 = 5.

6✏️ Worked examplePart 6 / 11

Rationalising the denominator

Convention requires no surd in a denominator, historically because dividing by an irrational number by hand was impractical.

Simple case. Multiply top and bottom by the surd.

3/√5 × √5/√5 = 3√5 / 5

Two-term denominator. Multiply by the conjugate, the same expression with the sign flipped.

1/(3 + √2) × (3 − √2)/(3 − √2)

The denominator becomes a difference of two squares:

(3 + √2)(3 − √2) = 9 − 2 = 7
= (3 − √2)/7

Why the conjugate works: the cross terms are +3√2 and −3√2, which cancel exactly, leaving only rational numbers.

7✏️ Worked examplePart 7 / 11

Standard form in calculations

Question. Evaluate (6 × 10⁵) ÷ (1.5 × 10⁻²), giving the answer in standard form.

Step 1: separate numbers and powers.

(6 ÷ 1.5) × (10⁵ ÷ 10⁻²)

Step 2: divide the numbers. 6 ÷ 1.5 = 4.

Step 3: subtract the indices. Careful with the double negative:

5 − (−2) = 7

Answer: 4 × 10⁷

Always check the form: the first number must satisfy 1 ≤ A < 10. If you had obtained 40 × 10⁶, it would need converting to 4 × 10⁷.

That final adjustment is where marks are most often dropped.

8🌍 In the real worldPart 8 / 11

Why this foundation matters

Everything builds on it. Quadratics produce surds; trigonometry produces exact values like √3/2; logarithms are index laws rearranged.

Exact versus approximate. Keeping √2 rather than 1.414 avoids rounding error accumulating through a long calculation. In engineering this matters.

Science. Standard form makes measurements from atomic to astronomical manageable and comparable.

The examination reality: questions asking for an "exact answer" require a surd, not a decimal. Rounding loses the mark even when the value is nearly right.

9⚠️ Common mistakesPart 9 / 11

Where marks are lost

√a + √b = √(a+b). Never.
Negative index makes it negative. It gives a reciprocal.
Taking the power before the root: correct but harder.
Leaving a surd in a denominator.
Answer outside 1-10 in standard form.
Rounding when "exact" was asked for.

10✅ SummaryPart 10 / 11

What to remember

Rational = terminating or recurring; √2 proved irrational by contradiction.
Root before power for fractional indices.
Only like surds add; rationalise using the conjugate.
"Exact" means leave the surd.

11✏️ Worked examplePart 11 / 11

Watch one worked through

Here is one done slowly, with the reason for every move. Then one for you.

Watch one first

Simplify (2³ × 2⁵) ÷ 2⁴, then evaluate 5⁻² and 7⁰.

Multiplying powers of the same base adds the indices.
2³ × 2⁵ = 2³⁺⁵ = 2⁸
It works because you are counting how many 2s are multiplied in total. The rule is a shortcut for counting, not a thing to memorise blindly.
Dividing subtracts the indices.
2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = 16
Dividing cancels pairs of 2s top and bottom, so the count goes down. The rules only apply when the base is the same.
A negative index means one over the positive power.
5⁻² = 1 ÷ 5² = 1/25
A negative index never makes the answer negative. It moves the term to the other side of the fraction line.
Anything to the power zero is 1.
7⁰ = 1
Because 7ⁿ ÷ 7ⁿ = 7⁰ and anything divided by itself is 1. Knowing why stops you writing 0.

Answer: 16, then 1/25, then 1

Now you try

Simplify 3⁴ × 3² ÷ 3⁵. Give the number.

Show the answer

Practise Real Numbers & Indices

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Real Numbers & Indices: Class 9 Mathematics Worksheet

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