Laws of exponents

Q: Why is any number to the power 0 equal to 1?

Because \dfrac{a^3}{a^3} = a^{3-3} = a^0, and any number divided by itself is 1.

Q: What does a negative exponent mean?

It means the reciprocal: a^{-n} = \dfrac{1}{a^n}. For example 10^{-2} = \dfrac{1}{100}.

Q: How do you write a very large number using exponents?

Use powers of ten: 4{,}500{,}000 = 4.5 \times 10^6.

An exponent says how many times a number (the base) is multiplied by itself: $2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32$ . The laws of exponents let you simplify expressions with powers without writing them out.

Formulas

Same base: multiply and divide

a^{m} \times a^{n} = a^{m + n} \frac{a ^{m}}{a ^{n}} = a^{m - n}

Power of a power, and of a product

(a^{m})^{n} = a^{mn} (ab)^{n} = a^{n} b^{n}

Zero and negative exponents

a^{0} = 1 a^{- n} = \frac{1}{a ^{n}}

Worked examples

Example 1. Simplify

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

Same base, so add the exponents: $2^{3 + 4} = 2^{7}$ .
$2^{7} = 128$ .

Answer: 128

Example 2. Simplify

5^{6} \div 5^{3}

Subtract the exponents: $5^{6 - 3} = 5^{3}$ .
$5^{3} = 125$ .

Answer: 125

Example 3. Simplify

(3^{2})^{3}

Multiply the exponents: $3^{2 \times 3} = 3^{6}$ .
$3^{6} = 729$ .

Answer: 729

Example 4. Find the value of

2^{- 3}

and

7^{0}

$2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ .
Any non-zero number to the power 0 is 1, so $7^{0} = 1$ .

Answer:

\frac{1}{8}

and 1

Common mistakes

Multiplying the bases: $2^{3} \times 2^{4}$ is $2^{7}$ , not $4^{7}$ . Only the exponents are added.
Using the laws on different bases. $2^{3} \times 3^{4}$ cannot be combined.
Thinking a negative exponent makes the number negative. $2^{- 3}$ is a small positive fraction, $\frac{1}{8}$ .

Practice questions with answers

1. Simplify:

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

A $2^{7}$
B $2^{12}$
C $4^{7}$
D $2^{1}$

Show answer

Answer: A)

2^{7}

When bases are equal, add the powers:

2^{3} \times 2^{4} = 2^{3 + 4} = 2^{7}

2. What is

5^{3}

A125
B15
C75
D243

Show answer

Answer: A) 125

5^{3}

means

5 \times 5 \times 5 = 125

3. Write

5^{- 2}

as a fraction.

A $\frac{1}{25}$
B $\frac{1}{10}$
C-25
D25

Show answer

Answer: A)

\frac{1}{25}

A negative exponent means reciprocal:

5^{- 2} = \frac{1}{5 ^{2}} = \frac{1}{25}

4. Simplify:

2^{3} * 2^{4}

A $2^{7}$
B $2^{12}$
C $4^{7}$
D $2^{1}$

Show answer

Answer: A)

2^{7}

Same base, add exponents:

2^{3} * 2^{4} = 2^{3 + 4} = 2^{7}

5. Anjali has to find the unknown power

n

in the equation:

(3^{n} x 3^{4}) / 3^{2} = 3^{9}

Work through the laws of exponents to find

n

Show answer

Answer: A) 7

Left side:

(3^{n} x 3^{4}) / 3^{2} = 3^{n + 4} / 3^{2} = 3^{n + 4 - 2} = 3^{n + 2}

. Setting

n + 2 = 9

gives

n = 7

6. Simplify

((- 5)^{2})^{1}

and give the value.

Show answer

Answer: C) 25

(- 5)^{2} = 25

, and raising 25 to the power 1 leaves it unchanged, so the value is 25.

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Frequently asked questions

What are the laws of exponents?

The six rules listed above: product, quotient, power of a power, power of a product, zero exponent and negative exponent.

Why is any number to the power 0 equal to 1?

Because $\frac{a ^{3}}{a ^{3}} = a^{3 - 3} = a^{0}$ , and any number divided by itself is 1.

What does a negative exponent mean?

It means the reciprocal: $a^{- n} = \frac{1}{a ^{n}}$ . For example $1 0^{- 2} = \frac{1}{100}$ .

How do you write a very large number using exponents?

Use powers of ten: $4, 500, 000 = 4.5 \times 1 0^{6}$ .

Laws of exponents

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

Related topics