AFCAT Numerical Ability · Indices and Surds

The Laws of Exponents

Multiply powers of one base by adding the indices, divide by subtracting, raise a power by multiplying, and read zero and negative powers correctly.

All Indices and Surds notes

Why this matters

Every later page leans on these few rules, so this page asks them directly: a product or quotient of powers, a zero or negative index, and a fraction of letters raised to a power. Most of it is EASY; the marks go on a sign slip, adding indices where they should be multiplied, or reading a negative power as a negative number.

Concept 1 of 3: Product, quotient and power of a power

a^m

a

multiplied by itself

m

times. Multiplying

a^m

a^n

just strings the factors together, so the indices add. Dividing cancels factors, so they subtract. Raising to a power repeats the whole block, so they multiply.

Definition

Product: $a^m \times a^n = a^{m+n}$ .
Quotient: $a^m \div a^n = a^{m-n}$ .
Power of a power: $(a^m)^n = a^{mn}$ .
Power of a product or quotient: $(ab)^n = a^n b^n$ and $\left(\dfrac ab\right)^n = \dfrac{a^n}{b^n}$ .
The first three need the same base. With two bases, sort the powers base by base and work on each one.

The three index laws

a^m a^n = a^{m+n}, \quad \dfrac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

Worked example

Simplify

\dfrac{5^6 \times 7^3}{5^4 \times 7^2}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Indices and Surds · Laws of Exponents

Simplify:

\frac{2^3 \times 3^4}{2^2 \times 3^2}

Add indices only on the same base

4^3 \times 5^3

(4 \times 5)^3 = 20^3

, not

20^6

. Indices add when the base is the same; when only the index is the same, the bases multiply and the index stays.

A power of a power multiplies

(7^2)^3 = 7^6

, but

7^2 \times 7^3 = 7^5

. Brackets mean multiply the indices; a product means add them.

Drill 6 more on product, quotient and power of a power

Concept 2 of 3: Zero and negative indices

Keep dividing by the base and watch the pattern:

3^2 = 9

3^1 = 3

3^0 = 1

3^{-1} = \tfrac 13

. A zero index is 1, and a negative index is the reciprocal. Nothing turns negative.

Definition

$a^0 = 1$ for any $a \ne 0$ .
$a^{-n} = \dfrac{1}{a^n}$ , and $\dfrac{1}{a^{-n}} = a^n$ .
A fraction to a negative power flips: $\left(\dfrac ab\right)^{-n} = \left(\dfrac ba\right)^{n}$ .
A decimal base is a fraction first: $0.2 = \tfrac 15$ , so $(0.2)^{-1} = 5$ .
The product and quotient laws still hold: $x^6 \times x^{-9} = x^{-3}$ .

Negative index

a^{-n} = \dfrac{1}{a^n}, \qquad \left(\dfrac ab\right)^{-n} = \left(\dfrac ba\right)^{n}

Worked example

Find

\left(\dfrac 25\right)^{-3}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Indices and Surds · Laws of Exponents

Find the value of:

(0.5)^{-2}

A negative index is not a negative number

3^{-3} = \tfrac{1}{27}

, not

-27

and not

-9

. The minus sign in the index means take the reciprocal.

A zero index gives 1, not 0

7^0 = 1

. The same holds for any non-zero base, and

x^{n} \times x^{-n} = x^0 = 1

Drill 7 more on zero and negative indices

Concept 3 of 3: Simplifying fractions of letters with powers

A fraction such as

\dfrac{p^4 q^{-1}}{p^{-4} q^3}

looks busy, but it is two small quotients side by side: one for

p

, one for

q

. Subtract the indices letter by letter, then write any negative index as a reciprocal.

Definition

Treat each letter on its own: top index minus bottom index.
Subtracting a negative index adds: $p^4 \div p^{-4} = p^{8}$ .
A bracket raised to a power multiplies every index inside it, coefficients included: $(2y^3)^4 = 16y^{12}$ .
Finish with positive indices only: a letter with a negative index moves below the line.

Letter by letter

\dfrac{p^a q^b}{p^c q^d} = p^{a-c}\, q^{b-d}

Worked example

Simplify

\dfrac{p^4 q^{-1}}{p^{-4} q^{3}}

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Indices and Surds · Laws of Exponents

Simplify:

\frac{a^5 \times b^{-3}}{a^{-2} \times b^2}

Minus a negative is plus

\dfrac{x^6}{x^{-3}}

the index is

6 - (-3) = 9

, not

3

. Write the subtraction out with brackets before you do it.

The coefficient takes the power too

\left(\dfrac{3m}{n}\right)^{-2} = \dfrac{n^2}{9m^2}

. Flipping the fraction is not enough: the 3 is squared along with

m

Drill 3 more on simplifying fractions of letters with powers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Product, quotient and power of a power
The three index laws
$a^m a^n = a^{m+n}, \quad \dfrac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}$
Zero and negative indices
Negative index
$a^{-n} = \dfrac{1}{a^n}, \qquad \left(\dfrac ab\right)^{-n} = \left(\dfrac ba\right)^{n}$
Simplifying fractions of letters with powers
Letter by letter
$\dfrac{p^a q^b}{p^c q^d} = p^{a-c}\, q^{b-d}$

Watch out for (6)

Add indices only on the same base
→ Product, quotient and power of a power
A power of a power multiplies
→ Product, quotient and power of a power
A negative index is not a negative number
→ Zero and negative indices
A zero index gives 1, not 0
→ Zero and negative indices
Minus a negative is plus
→ Simplifying fractions of letters with powers
The coefficient takes the power too
→ Simplifying fractions of letters with powers

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