Understand · explore · practise

Laws of indices

Simplify powers, understand negative and fractional indices, and keep track of the conditions that make each rule valid.

Before you startMultiplication, fractions and basic algebra

01 / The rules

A power counts repeated factors.

For a positive integer n, aⁿ means n copies of a multiplied together. The base is a; n is the index or exponent.

a^r × a^s = a^r+s

The groups of factors join together.

a^r ÷ a^s = a^r−s

Matching factors cancel; the base must not be zero.

(a^r)^s = a^rs

There are s groups, each containing r factors.

For integer powers these rules work with negative bases too, wherever the expressions are defined. When fractional powers enter, use a positive base for the rules on this page.

The operation matters: a² + a³ is a sum, not a product. You cannot add the exponents here.

Count the factorsExplore

First power m 2Second power n 4

3² × 3⁴ = 3⁶. Add the powers because you join two groups of factors.

Watch two groups of factors combine

Pause, replay or seek freely. The notes explain the same idea and stay in view.

02 / Zero & negative

Keep the pattern consistent.

Divide by the base each time the exponent falls by one. For base 5, this gives 125, 25, 5, 1, 1/5, 1/25 as the exponent goes from 3 down to −2.

a⁰ = 1 (a ≠ 0)

a⁻ⁿ = 1/aⁿ (a ≠ 0)

A negative exponent creates a reciprocal. It does not make the answer negative.

Brackets also matter: (−5)² = 25, but −5² = −25. In the second expression, the square is evaluated before the outside minus sign.

Do not use a⁰ = 1 to assign a value to 0⁰. It is excluded from this rule.

Why the reciprocal is neededWorked example

a³ ÷ a³ = 1

Any non-zero quantity divided by itself is 1.

a³⁻³ = a⁰ = 1

The index law must give the same result.

a² ÷ a⁵ = 1/a³

Cancel two factors of a from numerator and denominator.

a²⁻⁵ = a⁻³ = 1/a³

The negative power describes the remaining denominator.

03 / Fractional powers

The denominator names the root.

For a > 0 and positive integer n, the power 1/n means the positive nth root. Raising that root to the power p gives a^p/n.

a^1/n = ⁿ√a
a^p/n = (ⁿ√a)^p

For example, a power of 2/3 asks for a cube root and a square. Taking the root first usually keeps the arithmetic small.

A terminating decimal index can be written as a fraction: 0.75 = 3/4. Use the same index laws after converting it.

The square-root symbol gives the non-negative root: √36 = 6. Solving z² = 36 is a different task and gives z = ±6.

A root containing a variable

√(49x¹⁰) = (49x¹⁰)^1/2
= 7x⁵, x > 0

The square-root power applies to both factors. If x can be negative, the answer is 7|x⁵| instead.

What about negative bases?

Odd roots can be real: ∛(−27) = −3. Even roots of negative numbers are not real. To avoid invalid rewrites with fractional indices, the general fractional-power rules here assume a positive base.

Also, √(x²) = |x| for real x. It equals x only when x ≥ 0.

Root, power, reciprocalWorked example

216^−2/3 = 1/(216^2/3)

The minus sign in the exponent means reciprocal.

216^1/3 = 6

The cube root is 6 because 6³ = 216.

216^2/3 = 6² = 36

Now square the root.

216^−2/3 = 1/36

Keep the result exact.

04 / Mixed expressions

Separate the numbers from the powers.

Handle numerical coefficients, powers of x and powers of y separately. State any restrictions before cancelling a denominator.

(ab)ⁿ = aⁿbⁿ

The exponent applies to every factor inside the brackets. Thus (4x³)² = 16x⁶, not 4x⁶. For fractional n, take positive a and b.

In a sum over one denominator, divide each numerator term:

(12x⁵ + 8x²)/(4x²)
= 3x³ + 2, x ≠ 0

Different bases can be combined when the exponent matches: 2ⁿ × 7ⁿ = 14ⁿ. Different bases with different exponents have no corresponding shortcut.

Combine roots, brackets and division

(x + √x)²/x
= (x² + 2x√x + x)/x
= x + 2√x + 1, x > 0

Expand the numerator before dividing each term by x. The middle term does not disappear.

A second example with an outside power

(4x^−3/2)²/(8x⁻¹)
= 16x⁻³/(8x⁻¹)
= 2x⁻² = 2/x²

Here x > 0. Square the coefficient and multiply the exponent by 2 before using the quotient rule.

One variable at a timeWorked example

(18x^7/3y⁻¹)/(6x^1/3y²)

Assume x > 0 and y ≠ 0.

= 3x^7/3−1/3y⁻¹⁻²

Divide 18 by 6. Subtract each denominator exponent.

= 3x²y⁻³

7/3 − 1/3 = 2; −1 − 2 = −3.

= 3x²/y³

Write with positive indices if requested.

05 / Your turn

Check the reasoning as well as the answer.

Try these on paper. Keep fractional powers exact and write the rule you use between each pair of lines.

Unless a question says otherwise, take all variables in this practice to be positive.

01 · Simplify

Write with positive indices.

(12a⁵b⁻²)/(3a⁻¹b)

Hint

Work on the coefficient, a and b separately. Subtract the whole denominator exponent.

Worked solution

= 4a⁵⁻⁽⁻¹⁾b⁻²⁻¹
= 4a⁶b⁻³
= 4a⁶/b³

02 · Keep it exact

Evaluate without converting to a decimal.

(16/81)^−3/4

Hint

Find the fourth root of 16/81 first. Then cube it and take the reciprocal.

Worked solution

(16/81)^1/4 = 2/3
(2/3)³ = 8/27
(16/81)^−3/4 = 27/8

03 · Find the missing power

Find p so this identity holds for every t > 0.

(t^p × t^1/2)/t⁻³ = t⁶

Hint

The combined exponent on the left must be 6.

Worked solution

p + 1/2 − (−3) = 6
p = 5/2

The condition is “for every t”. At t = 1 alone, any exponent would give 1.

04 · Use a substitution

Given y = 3x⁻², express y²/(6x⁻¹) as kxⁿ.

Hint

Substitute the entire expression for y inside brackets before squaring.

Worked solution

y²/(6x⁻¹)
= (3x⁻²)²/(6x⁻¹)
= 9x⁻⁴/(6x⁻¹)
= (3/2)x⁻³

So k = 3/2 and n = −3.

05 · Diagnose a mistake

A student says (x³)⁴ = x⁷ because 3 + 4 = 7. Explain the mistake and give a numerical check.

Hint

Write (x³)⁴ as four copies of x³.

Worked solution

(x³)⁴ = x³ × x³ × x³ × x³
= x¹²

At x = 2, the original is 8⁴ = 4096. The proposed x⁷ gives 128. Adding exponents belongs to multiplication of powers, not a power raised to another power.

06 · Same base, unknown exponent

Solve 125^2t−1 = 5^t+6.

Hint

Write 125 as 5³. Then compare the exponents of the same positive base, 5.

Worked solution

5^3(2t−1) = 5^t+6
6t − 3 = t + 6
5t = 9
t = 9/5

Both sides then have exponent 39/5 when written as powers of 5.

06 / Recap

Choose the rule by the operation.

Multiply same-base powers: add exponents.
Divide same-base powers: subtract exponents; do not divide by zero.
Raise a power to a power: multiply exponents.
Negative exponent: take a reciprocal. Fractional exponent: take a root and a power.
Brackets matter: the power applies to every factor inside them.

A quick numerical substitution can expose a mistake. It does not, on its own, prove an identity for every value.

Next: expanding and factorising →

Section 1 of 6 · The rules