x² Indices Laws — Algebra 2

An index (power or exponent) tells you how many times a number multiplies itself. There are 5 laws that let you simplify any expression involving powers. Engineers use them in stress calculations, electrical formulas, area and volume, and standard form.

5 core lawsFractional indices = roots Negative indices = reciprocalsEngineering standard form

index, power or exponent. The big number (2) is called the base.

Key VocabularyParts of an index expression

x⁵ → base = x index = 5

3⁴ → base = 3 index = 4 → 81

a² → base = a index = 2 → a×a

Why indices matter in engineering

Area — a square of side x has area x². All area formulas involve squaring.
Volume — a cube of side x has volume x³. Cubic relationships everywhere.
Standard form — 2.5 × 10⁵ uses index 5. Used for very large/small engineering values.
Power formulas — P = I²R uses x². KE = ½mv² also uses squaring.
Simplifying — the 5 laws let you simplify complex expressions in one step.

💡 Indices are so common in engineering that every formula sheet you’ll ever use contains them. Learn the 5 laws and you can handle all of them!

See the Power2¹, 2², 2³, 2⁴ — see how fast powers grow

📌 This exponential growth is why engineers use standard form — values like 1.6 × 10²⁰ are impossible to write out normally. Powers make huge and tiny numbers manageable.

×

Law 1 — Multiplying Same Base: Add the Powers

When you multiply two powers with the same base, you simply add the indices. Think of it as combining groups of multiplications.

LAW 1

x^m × xⁿ = x^m+n

When multiplying powers with the same base, ADD the indices.

x³ × x² = x⁵ (3+2=5)
a⁴ × a⁶ = a¹⁰ (4+6=10)
2³ × 2² = 2⁵ = 32 (check: 8×4=32 ✔)

Why adding worksThe reason behind the rule

Write it out in full to see why:

x³ × x² = (x×x×x) × (x×x)

= x×x×x×x×x = x⁵

✅ You’re just counting: 3 lots of x multiplied by 2 lots of x = 5 lots of x multiplied together.

⚠️ CRITICAL: Law 1 ONLY works when both powers have the same base!
x³ × y² ≠ (xy)⁵ — different bases, cannot combine!

Worked Examples

Simplify: a⁴ × a⁹

Same base? Yes — both base a

Add powers: 4 + 9 = 13

Answer: a¹³ ✅

Simplify: 3x² × 5x⁴

Coefficients: 3 × 5 = 15

Powers of x: 2 + 4 = 6

Answer: 15x⁶ ✅

÷

Law 2 — Dividing Same Base: Subtract the Powers

When you divide two powers with the same base, subtract the index on the bottom from the index on top. The cancelling of x’s shows exactly why this works.

LAW 2

x^m ÷ xⁿ = x^m−n

When dividing powers with the same base, SUBTRACT the indices (top minus bottom).

x⁵ ÷ x² = x³ (5−2=3)
a⁹ ÷ a⁴ = a⁵ (9−4=5)
2⁵ ÷ 2³ = 2² = 4 (check: 32÷8=4 ✔)

Why subtracting worksThe cancellation proof

x⁵ ÷ x² = x·x·x·x·x / x·x

= x·x·x·❌·❌ / ❌·❌

= x³

✅ The x·x on the bottom cancels two of the x’s on top — leaving exactly x³.

⚠️ Subtract top minus bottom — not the other way!
x⁵ ÷ x² = x³ (5−2) NOT x⁻³ (2−5)

Worked Examples

Simplify: y¹⁰ ÷ y⁴

Same base? Yes — both y

Subtract: 10 − 4 = 6

Answer: y⁶ ✅

Simplify: 12x⁸ ÷ 4x⁵

Coefficients: 12 ÷ 4 = 3

Powers: 8 − 5 = 3

Answer: 3x³ ✅

²

Law 3 — Power of a Power: Multiply the Indices

When a power is raised to another power — i.e. a bracket with a power outside — you multiply the two indices together.

LAW 3

(x^m)ⁿ = x^m×n

When raising a power to another power, MULTIPLY the indices.

(x³)² = x⁶ (3×2=6)
(a²)⁴ = a⁸ (2×4=8)
(2³)² = 2⁶ = 64 (check: 8²=64 ✔)

With coefficientsDon’t forget to raise the coefficient too!

⚠️ When expanding (2x³)², the power applies to everything inside the bracket — both the number and the variable!

(2x³)² = 2² × (x³)² = 4x⁶

(3a²)³ = 3³ × (a²)³ = 27a⁶

✅ Step 1: raise the coefficient to the power.
Step 2: multiply the index of the variable by the power.

Worked Examples

Simplify: (y²)⁶

Multiply: 2 × 6 = 12

Answer: y¹² ✅

Simplify: (4x³)²

Coefficient: 4² = 16

Variable: (x³)² = x⁶ (3×2=6)

Answer: 16x⁶ ✅

∅

Laws 4 & 5 — Zero Power & Negative Indices

These two laws surprise most students at first — but they flow naturally from Law 2. Once you see the proof, they make perfect sense.

LAW 4

x⁰ = 1

Any base raised to the power zero equals exactly 1. Always. No exceptions (except 0⁰ which is undefined).

5⁰ = 1 x⁰ = 1 1000⁰ = 1
(−7)⁰ = 1 (a+b)⁰ = 1

Proof using Law 2

x³ ÷ x³ = x³⁻³ = x⁰ (by Law 2)

But x³ ÷ x³ = 1 (anything ÷ itself = 1)

Therefore x⁰ = 1 ✅

💡 Anything divided by itself = 1. Law 2 gives us x⁰ for that case. So x⁰ must equal 1!

LAW 5

x⁻ⁿ = 1 / xⁿ

A negative index means the reciprocal (flip it). The negative sign means “move it to the other side of the fraction bar”.

x⁻² = 1/x² x⁻⁵ = 1/x⁵
2⁻³ = 1/2³ = 1/8 5⁻² = 1/25

Proof using Law 2

x² ÷ x⁵ = x²⁻⁵ = x⁻³ (by Law 2)

But x² ÷ x⁵ = x·x / x·x·x·x·x = 1/x³

Therefore x⁻³ = 1/x³ ✅

⚠️ x⁻² does NOT mean “negative x squared”!
x⁻² = 1/x² (positive result if x is real)

VisualNegative index = flip the fraction

√

Fractional Indices — Powers as Roots

A fractional index means a root. The denominator of the fraction tells you which root to take. This connects indices directly to square roots, cube roots and beyond.

FRACTIONAL INDICES

x¹ᐟⁿ = ᵞ√x x^mᐟⁿ = ᵞ√(x^m)

Bottom of fraction = which root | Top of fraction = power to raise to

x¹ᐟ² = √x x¹ᐟ³ = ∛x x²ᐟ³ = (∛x)²
16¹ᐟ² = √16 = 4 27¹ᐟ³ = ∛27 = 3
8²ᐟ³ = (∛8)² = 2² = 4

Proof from Law 3Why x¹ᐟ² = √x

Law 3: (x¹ᐟ²)² = x¹ᐟ² × ² = x¹ = x

So x¹ᐟ² is a number that, squared, gives x

That is exactly the definition of √x!

Therefore x¹ᐟ² = √x ✅

💡 Denominator = root number.
x¹ᐟ² = square root x¹ᐟ³ = cube root x¹ᐟ⁴ = 4th root

Step-by-step method for x^mᐟⁿ

Take the root first (easier with smaller numbers)
Then raise to the power

Evaluate: 32³ᐟ⁵

Root first: ⁵√32 = 2 (since 2⁵=32)

Then power: 2³ = 8

Answer: 8 ✅

Evaluate: 64²ᐟ³

Root first: ∛64 = 4

Then power: 4² = 16

Answer: 16 ✅

📋

All 5 Laws — Quick Reference Card

Everything on one page. Memorise these and you can handle any indices question in the T Level exam.

Law 1 — Multiply x^m × xⁿ = x^m+n x³ × x² = x⁵

Law 2 — Divide x^m ÷ xⁿ = x^m−n x⁵ ÷ x² = x³

Law 3 — Power of Power (x^m)ⁿ = x^m×n (x²)³ = x⁶

Law 4 — Zero Index x⁰ = 1 5⁰ = 1 x⁰ = 1

Law 5 — Negative Index x⁻ⁿ = 1/xⁿ x⁻² = 1/x²

Common Mistakes Summary

❌ x² × x³ = x⁶ — WRONG (don’t multiply powers in Law 1!)

❌ (x²)³ = x⁵ — WRONG (don’t add powers in Law 3!)

❌ x⁰ = 0 — WRONG (zero power = 1, not 0!)

❌ x⁻² = −x² — WRONG (negative index = reciprocal, not negative!)

❌ x³ × y² = (xy)⁵ — WRONG (Laws only work with the SAME base!)

Quick Memory Tricks

💡 Law 1: “Multiply = Add” (operations opposite)

💡 Law 2: “Divide = Subtract” (operations opposite)

💡 Law 3: “Power of power = Multiply indices”

💡 Law 4: “Zero power = 1 always”

💡 Law 5: “Negative power = flip it”

💡 Fractional: “Bottom = root, Top = power”

⚙️

Applications

Where this topic is used in engineering, manufacturing, maintenance and daily life.

⚡Electrical: P=I²R. Power involves squared current.
💾Technology: 1 KB=2¹⁰ bytes, 1 MB=2²⁰. Scientific notation.
🔬Science: Avogadro: 6.022×10²³. Atom radius: 1.2×10⁻¹⁰ m.
⚙️Mechanical: σ=F/A. A=4×10⁻³ m². Use index laws to simplify.

⚙

Indices in Engineering

Every engineering formula involving area, volume, power or standard form uses indices. Here’s where each law actually appears in T Level / Level 3 Engineering work.

⚡ Electrical Power — P = I²R

Power formula uses squaring. When simplifying circuits with multiple components, you need Law 1 and Law 2 to combine expressions.

P = I²R (Law: squaring)
If I = 3A, R = 4Ω: P = 3²×4 = 36W

Simplify: I² × I³ = I⁵
Applied: total current term in series circuit

🔨 Stress — σ = F/A, Area = πr²

Area formulas for circular cross-sections use x². When r doubles, area increases by 2² = 4 times (Law 3 in action).

A = πr²
If r doubles to 2r: A = π(2r)² = 4πr²
Area is 4 times bigger (Law 3: (2r)²)

📏 Standard Form — 10^m Laws

All standard form calculations use Law 1 and Law 2 with base 10. Multiplying numbers in standard form means adding powers of 10.

(3×10²) × (4×10⁴)
= 3×4 × 10²⁺⁴ (Law 1)
= 12 × 10⁶
= 1.2 × 10⁷

🌡 Thermal Expansion — Volume

Volume of a cube = s³. Small changes in side length cause cubic changes in volume. Uses Law 3 when expanding (1+αᵇ)³.

V = s³ (cube of side)
If s doubles: V = (2s)³ = 8s³
Volume 8 times bigger (Law 3)

🔧 T Level Exam Style — Simplify then evaluate

Problem: Simplify (2x³)² × x⁻², then evaluate when x = 3

Law 3: (2x³)² = 2² × x⁶ = 4x⁶

Law 1: 4x⁶ × x⁻² = 4x⁶⁻² = 4x⁴

Evaluate: 4(3)⁴ = 4 × 81

Answer: 4x⁴ | value = 324 ✅

🔢

Interactive Indices Tool

Explore every index law with step-by-step working, and evaluate any power expression. Change the values and watch the working update instantly.

📐 Indices Law ExplorerAll 7 laws with full working

ℹ️ How to use: Type a base and two powers into the boxes, then use the dropdown to choose which law to explore. Every step of the working updates automatically as you type.

Base (x)

Power m

Power n

Choose a law

Select a law from the dropdown above to see the step-by-step working.

👆 Try these: Law 1: 2³×2⁴ · Law 2: 5⁴÷5² · Law 3: (3²)⁴ · Frac: 8²ᐟ³

🧮 Evaluate x^m — Any Base & PowerDecimal, fraction & negative powers

ℹ️ How to use: Type any base number and any power (including decimals like 0.5, negatives like −2, or fractions like 2/3). Click Evaluate to see the result and which law applies.

Base

^

Power (e.g. 0.5, −2, 2/3)

Enter a base and power above, then click Evaluate.

👆 Try: 4^0.5 = √4 · 8^(2/3) · 2^−3 = 1/8 · 7^0 = 1

All Laws at a Glance

Law 1 x^m × xⁿ = x^m+n

Law 2 x^m ÷ xⁿ = x^m−n

Law 3 (x^m)ⁿ = x^m×n

Law 4 x⁰ = 1

Law 5 x⁻ⁿ = 1/xⁿ

Frac x^mᐟⁿ = (ⁿ√x)^m

Common mistakes to avoid

⚠ x⁰ ≠ 0 — anything to the power zero equals 1.

⚠ x⁻² ≠ −x² — negative index means reciprocal, not negative.

⚠ (x+y)ⁿ ≠ xⁿ+yⁿ — Law 1 only works when the base is the same.

✅ Always check: same base? Then apply the laws. Different bases? Evaluate separately first.

🎮

Quick Fire Quiz

Test your knowledge — 10 questions, instant feedback.

⚡ Indices Blitz

ℹ️ How to play: An index expression appears on screen. Click the correct value from the four options. Your score and streak update after each question — click Next to continue.

Score: 0 Streak: 0 🔥 Q 1/10

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✏

Practice Questions

15 questions across 3 tiers. Attempt each one before revealing the answer. State which law you’re using!

🟢 Tier 1 — Foundations

ℹ️ How to use: Work through each question and write down your answer. When ready, click Show Answer to reveal the full worked solution and mark scheme.

Q1

Write out 3⁴ as a multiplication and calculate the value.

[2]

3⁴ = 3×3×3×3 = 81

Q2

Simplify: x⁵ × x³

[1]

Which law? Add or subtract?

Law 1 (multiply → add powers)
5 + 3 = 8
x⁸

Q3

Simplify: y¹⁰ ÷ y⁴

[1]

Law 2 (divide → subtract powers)
10 − 4 = 6
y⁶

Q4

Evaluate: 7⁰

[1]

Law 4 (zero power = 1)
7⁰ = 1

Q5

Write as a fraction: x⁻⁴

[1]

Law 5 (negative index → reciprocal)
1/x⁴

🟡 Tier 2 — Core Skills

Q6

Simplify: (x³)⁴

[1]

Law 3 (power of power → multiply)
3 × 4 = 12
x¹²

Q7

Simplify: 4x² × 3x⁵

[2]

Coefficients: 4×3=12
Powers (Law 1): 2+5=7
12x⁷

Q8

Simplify: 15a⁸ ÷ 3a⁵

[2]

Coefficients: 15÷3=5
Powers (Law 2): 8−5=3
5a³

Q9

Simplify: (3x²)³

[2]

Coefficient: 3³=27
Variable (Law 3): (x²)³=x⁶
27x⁶

Q10

Evaluate: 64¹ᐟ㎥ (i.e. 64 to the power one-third)

[2]

Denominator = cube root

64¹ᐟ³ = ∛64
2³=8, 3³=27, 4³=64 → answer = 4

🔴 Tier 3 — T Level Challenge

Q11

Simplify fully: x³ × x⁻⁵ × x²

[2]

Laws 1&5: add all powers: 3+(−5)+2=0
x⁰ = 1

Q12

Simplify: (2a²b³)² ÷ (4a³b)

[4]

Expand top (Law 3): 4a⁴b⁶
Divide (Law 2): 4a⁴b⁶ ÷ 4a³b
Coefficients: 4÷4=1
a: 4−3=1 → a b: 6−1=5 → b⁵
ab⁵

Q13

Evaluate 27²ᐟ³ without a calculator. Show all steps.

[3]

Step 1 (root): ∛27 = 3
Step 2 (power): 3² = 9

Q14

Using P = I²R, if I = 2x and R = 3x², write an expression for P in simplest index form.

[3]

P = (2x)² × 3x²
= 4x² × 3x² (Law 3)
= 12x⁴ (Law 1)
P = 12x⁴

Q15

Simplify: (x³y²)² × (xy³)⁻². Write the answer using positive indices only.

[5]

Expand (x³y²)² = x⁶y⁴ (Law 3)
Expand (xy³)⁻² = x⁻²y⁻⁶ (Laws 3&5)
Multiply: x⁶·x⁻² × y⁴·y⁻⁶ (Law 1)
x: 6−2=4 → x⁴
y: 4−6=−2 → y⁻² = 1/y²
x⁴/y²

🎯 Score guide: Q1–5 = foundations (7 marks) · Q6–10 = core skills (9 marks) · Q11–15 = T Level challenge (17 marks)

👉 Next topic: Expanding Brackets — using the laws you’ve just learned to multiply out single and double brackets.

x² Indices Laws — Algebra 2

Law 1 — Multiplying Same Base: Add the Powers

Law 2 — Dividing Same Base: Subtract the Powers

Law 3 — Power of a Power: Multiply the Indices

Laws 4 & 5 — Zero Power & Negative Indices

Fractional Indices — Powers as Roots

All 5 Laws — Quick Reference Card

Applications

Indices in Engineering

⚡ Electrical Power — P = I²R

🔨 Stress — σ = F/A, Area = πr²

📏 Standard Form — 10m Laws

🌡 Thermal Expansion — Volume

Interactive Indices Tool

Quick Fire Quiz

⚡ Indices Blitz

Practice Questions

📏 Standard Form — 10^m Laws