MTH1W The Exponent Laws: Notes and Practice

A thousand is $1 0^{3}$ and ten thousand is $1 0^{4}$ . What is a thousand times ten thousand? It is ten million, 10 000 000, which is $1 0^{7}$ . Notice that $3 + 4 = 7$ . That is not a coincidence. When you multiply powers of the same base, you are just joining two strings of factors into one longer string, so the exponents (the counts of factors) add. The exponent laws are three shortcuts like this one. You do not need to memorize them blindly: each one comes straight from writing the factors out, and if you ever forget a law you can rebuild it in ten seconds by expanding a small example.

The product law: multiply powers, add exponents

Fig 1Factor blocks for three laws: a cubed times a squared joins 3 and 2 blocks into 5; a to the 5 divided by a squared cancels 2 pairs to leave 3; a squared, cubed, makes 3 groups of 2 blocks, 6 in all.

2^{3} \times 2^{2} = (2 \times 2 \times 2) \times (2 \times 2) = 2^{5} ⟹ a^{m} \times a^{n} = a^{m + n}

The product law says: when you multiply powers with the same base, keep the base and add the exponents. The base does not change: $2^{3} \times 2^{2}$ is $2^{5}$ , not $4^{5}$ , because every factor is still a 2. The law needs a common base. In $2^{3} \times 3^{2}$ the factors are different numbers, so there is no shortcut; you evaluate: $8 \times 9 = 72$ . With variables, numbers in front (coefficients) multiply as usual while exponents on the same variable add: $3 x^{2} \times 4 x^{5} = (3 \times 4) (x^{2} \times x^{5}) = 12 x^{7} .$

Quick checkTry it before you read on

Simplify

5^{3} \times 5^{4}

Quick checkTry it before you read on

Simplify

2 x^{3} \times 5 x^{4}

. Type powers with ^: type 3x^2 for

3 x^{2}

The quotient law: divide powers, subtract exponents

\frac{a ^{5}}{a ^{2}} = \frac{a \times a \times a \times a \times a}{a \times a} = a^{3} ⟹ \frac{a ^{m}}{a ^{n}} = a^{m - n} (a \neq = 0)

Each factor on the bottom cancels one matching factor on the top, so dividing by $a^{2}$ removes 2 factors: subtract the exponents. The base cannot be 0, since you cannot divide by 0. The quotient law also explains the patterns from the previous lesson. Dividing a power by itself gives 1, and the law gives $\frac{a ^{3}}{a ^{3}} = a^{3 - 3} = a^{0}$ , so $a^{0} = 1$ . When the bottom exponent is larger, $\frac{a ^{2}}{a ^{5}}$ leaves three factors on the bottom, which is $\frac{1}{a ^{3}}$ , and the law gives $a^{2 - 5} = a^{- 3}$ . Both meanings agree, which is why the definitions of zero and negative exponents make sense.

Fig 2Factor blocks for a³ ÷ a³, where every factor cancels to leave 1 = a⁰, and for a² ÷ a⁵, where 2 pairs cancel and 3 factors stay on the bottom to leave 1/a³ = a⁻³.

Quick checkTry it before you read on

Write

a^{4} \div a^{6}

as a single power of

a

(with

a \neq = 0

Quick checkTry it before you read on

Evaluate

\frac{1 0 ^{6}}{1 0 ^{4}}

Power of a power, and power of a product

(a^{m})^{n} = a^{m \times n} (ab)^{n} = a^{n} b^{n}

Raising a power to a power repeats the whole power as a factor: $(2^{3})^{2} = 2^{3} \times 2^{3} = 2^{3 + 3} = 2^{6} .$ That is $2$ groups of $3$ factors, so multiply the exponents. When a product is raised to a power, every factor inside the brackets gets the exponent: $(2 x)^{3} = (2 x) (2 x) (2 x) = 2^{3} x^{3} = 8 x^{3} .$ Forgetting the coefficient is a very common slip: $(2 x)^{3}$ is not $2 x^{3}$ . One warning: the power-of-a-product law works for products only, not sums. $(2 + 3)^{2} = 5^{2} = 25$ , but $2^{2} + 3^{2} = 4 + 9 = 13$ , so $(a + b)^{2}$ is generally not $a^{2} + b^{2}$ .

Fig 3Top: (2x)³ as three groups of a 2 block and an x block, regrouped into 2 · 2 · 2 = 8 and x · x · x = x³, so (2x)³ = 8x³. Bottom: a 5 by 5 square split into a 4, a 9 and two 6s, showing (2 + 3)² = 25 while 2² + 3² = 13.

Law	Rule	Numeric example	Algebraic example
Product	$a^{m} \times a^{n} = a^{m + n}$	$5^{2} \times 5^{4} = 5^{6}$	$x^{3} \cdot x^{7} = x^{10}$
Quotient	$a^{m} \div a^{n} = a^{m - n}$ , $a \neq = 0$	$7^{8} \div 7^{3} = 7^{5}$	$\frac{y ^{9}}{y ^{4}} = y^{5}$
Power of a power	$(a^{m})^{n} = a^{mn}$	$(3^{2})^{4} = 3^{8}$	$(x^{4})^{3} = x^{12}$
Power of a product	$(ab)^{n} = a^{n} b^{n}$	$(2 \times 5)^{3} = 2^{3} \times 5^{3}$	$(3 x)^{2} = 9 x^{2}$
Zero exponent	$a^{0} = 1$ , $a \neq = 0$	$9^{0} = 1$	$(4 x)^{0} = 1$ when $x \neq = 0$
Negative exponent	$a^{- n} = \frac{1}{a ^{n}}$ , $a \neq = 0$	$4^{- 2} = \frac{1}{16}$	$x^{- 3} = \frac{1}{x ^{3}}$

The exponent laws with a numeric and an algebraic example of each.

The laws are for multiplying and dividing, not adding

x^{2} + x^{3}

cannot be simplified to

x^{5}

: there is no law for adding powers. Check with

x = 2

4 + 8 = 12

, but

2^{5} = 32

. Likewise

2^{3} + 2^{3} = 8 + 8 = 16 = 2^{4}

, not

2^{6}

. Only like terms with the same exponent can be added, as in

3 x^{2} + 5 x^{2} = 8 x^{2}

Quick checkTry it before you read on

Simplify

(x^{4})^{2}

Quick checkTry it before you read on

Simplify

(5 x)^{2}

VideoJensenMath, an Ontario teacher, works through the product, quotient and power of a power laws in a Grade 9 lesson. Watch it as a review of the three laws together.Watch on YouTube

Simplifying step by step

Deal with brackets raised to a power first, using the power-of-a-power and power-of-a-product laws.
Multiply the coefficients (the plain numbers) as usual.
For each variable separately, add exponents when multiplying and subtract when dividing.
Write any power with exponent 1 without the exponent ( $x^{1} = x$ ) and replace $x^{0}$ with 1 (for $x \neq = 0$ ).
If the question asks for a number, evaluate the final power at the end, when it is smallest.

Check by substituting a number

Pick a small value such as

x = 2

and evaluate the original expression and your answer. If they disagree, there is an error. For

(2 x^{3})^{2} = 4 x^{6}

: at

x = 2

the left side is

(2 \times 8)^{2} = 256

and the right side is

4 \times 64 = 256

. Avoid

x = 0

and

x = 1

for checks, since many wrong answers also agree there.

Quick checkTry it before you read on

You simplify

(x^{2})^{3}

and want to check your answer by substituting a number. Which value of

x

is the best choice?

Quick checkTry it before you read on

Simplify

(x^{3})^{2} \cdot x

. Type powers with ^: type x^3 for

x^{3}

The Exponent Laws

Key things to remember

The product law: multiply powers, add exponents

The quotient law: divide powers, subtract exponents

Power of a power, and power of a product

Simplifying step by step

Worked examples

Where marks get lost

Flashcards 7

Practice