Exponent laws - Year 8 Mathematics

Start here: why do we need laws?

Look at this: $2^3 \times 2^2$ . Instead of computing each power and then multiplying, what if we noticed this is really

\underbrace{(2 \times 2 \times 2)}_{2^3} \times \underbrace{(2 \times 2)}_{2^2} = 2 \times 2 \times 2 \times 2 \times 2 = 2^5.

Three twos, times two twos, makes five twos. The exponents add. That’s the whole idea behind the index laws — they are shortcuts that just count the factors correctly.

What you will learn

multiply powers with the same base by adding the exponents,
divide powers with the same base by subtracting the exponents,
raise a power to a power by multiplying the exponents,
use the zero exponent rule: $a^0 = 1$ for any non-zero $a$ ,
combine these laws to simplify compound expressions.

1. Recap of index notation

$a^n$ means $a$ multiplied by itself $n$ times: $3^4 = 3 \times 3 \times 3 \times 3 = 81$ . The base is $a$ , the exponent (or index) is $n$ .

Worked example E Very easy: count the factors

Simplify $\;x^2 \times x^3$ .

Expand both sides to see what’s happening:

x^2 \times x^3 = (x \cdot x) \times (x \cdot x \cdot x) = x^5.

Five factors of $x$ . The exponents $2$ and $3$ added to give $5$ .

2. The three index laws

Index laws (positive integer exponents, same base)

Product of powers

a^m \times a^n \;=\; a^{m + n}.

Example: $2^3 \times 2^5 = 2^{8} = 256$ .

Quotient of powers

a^m \div a^n \;=\; a^{m - n}, \quad a \neq 0.

Example: $5^7 \div 5^3 = 5^4 = 625$ .

Power of a power

(a^m)^n \;=\; a^{m \times n}.

Example: $(3^2)^4 = 3^8 = 6561$ .

Worked example 1 Product of powers

Simplify $\;x^4 \times x^7$ .

Same base, so add the exponents:

x^4 \times x^7 \;=\; x^{4 + 7} \;=\; x^{11}.

Worked example 2 Quotient of powers

Simplify $\;\dfrac{y^{12}}{y^{5}}$ .

Same base, subtract the exponents:

\dfrac{y^{12}}{y^{5}} \;=\; y^{12 - 5} \;=\; y^{7}.

Worked example 3 Power of a power

Simplify $\;(m^3)^{4}$ .

Multiply the exponents:

(m^3)^{4} \;=\; m^{3 \times 4} \;=\; m^{12}.

3. The zero exponent

What should $a^0$ equal? Use the quotient rule:

\dfrac{a^n}{a^n} \;=\; a^{n - n} \;=\; a^{0}.

But also $\dfrac{a^n}{a^n} = 1$ for any $a \neq 0$ . So both answers must agree:

Zero exponent rule

a^0 \;=\; 1, \qquad a \neq 0.

Examples: $7^0 = 1$ , $(-5)^0 = 1$ , $x^0 = 1$ (as long as $x \neq 0$ ).

4. Combining the laws

Worked example 4 Combining laws

Simplify $\;\dfrac{(a^3)^2 \times a^5}{a^4}$ .

Apply power of a power to $(a^3)^2 = a^6$ .
Product of powers on the top: $a^6 \times a^5 = a^{11}$ .
Quotient of powers: $\dfrac{a^{11}}{a^4} = a^{7}$ .

Final: $a^7$ .

Worked example 5 With numeric coefficients

Simplify $\;6x^5 \times 4x^3$ .

Multiply the numbers separately from the variables:

6x^5 \times 4x^3 \;=\; (6 \times 4) \times (x^5 \times x^3) \;=\; 24 x^{8}.

Practice: Year 8 core

Fluency

Apply one law at a time

Simplify $2^3 \times 2^6$ .
Simplify $a^5 \times a^2$ .
Simplify $x^{10} \div x^4$ .
Simplify $\dfrac{m^8}{m^3}$ .
Simplify $(3^2)^3$ .
Simplify $(p^4)^5$ .
Evaluate $7^0$ .
Evaluate $(-3)^0$ .
Simplify $y^{12} \div y^{12}$ .
Simplify $(x y^2)^3$ . (Hint: each factor raised to the power.)

Fluency

Combine the laws

Simplify $\dfrac{a^5 \times a^3}{a^4}$ .
Simplify $\dfrac{(b^3)^2}{b^{5}}$ .
Simplify $(2m)^3 \times m^2$ .
Simplify $\dfrac{10 x^7}{2 x^4}$ .
Simplify $5 a^3 \times 2 a^4$ .
Simplify $(3 x^2)^3$ .
Simplify $\dfrac{a^4 b^6}{a b^2}$ .
Simplify $\dfrac{(x^3)^4 \cdot x^2}{x^{10}}$ .

Reasoning

Explain and spot the mistake

Sam writes $2^3 \times 2^3 = 2^9$ . Is Sam correct? If not, what should it be?
Explain why $a^0 = 1$ is forced by the quotient rule.
Mira writes $(a^2)^3 = a^5$ . Explain the error and give the correct value.
Is $x^{-2}$ a Year 8 result, and if not, what does it mean? (Scope note: negative exponents appear in Year 9; for now think about what the quotient rule would give you.)

Problem solving

Applications

A bacterium doubles every hour. Starting from $1$ cell, write the number of cells after $n$ hours as a power of $2$ . How many cells after $8$ hours?
A cube of side $s$ has volume $s^3$ . Express the volume of a cube whose side is doubled, as a multiple of $s^3$ .
Computer memory is measured in powers of $2$ . How much bigger is $2^{20}$ than $2^{10}$ ? Express as a power of $2$ .
A square tile pattern has $n$ rows and $n$ columns. If each tile is $10$ cm by $10$ cm, express the total floor area in cm² using exponent notation.

Challenge

Reasoning

Harder reasoning

Simplify $\dfrac{(2 x^3 y)^2 \times (x y^2)^3}{x^4 y^5}$ .
If $a^m = 8$ and $a^n = 4$ , find $a^{m + n}$ without finding $a$ , $m$ or $n$ .
A formula for a population model is $P = P_0 \times 2^t$ where $P_0$ is the starting population and $t$ is time in years. If a town starts at $5000$ and the population doubles every year, what is the population after $6$ years?
Simplify and write as a power of $2$ : $\dfrac{8^5 \times 4^3}{32^2}$ . (Hint: write every base as a power of $2$ .)

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

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Year 8 core - answers

Fluency

Apply one law at a time

$2^9 = 512$
$a^7$
$x^6$
$m^5$
$3^6 = 729$
$p^{20}$
$1$
$1$
$1$ (by quotient rule the exponent is $0$ )
$x^3 y^6$ . Method: raise each factor to the power $3$ .

Fluency

Combine the laws

$a^4$ . Method: top $a^8$ ; $a^8 \div a^4 = a^4$ .
$b$ . Method: top $b^6$ ; $b^6 \div b^5 = b^1 = b$ .
$8 m^5$ . Method: $(2m)^3 = 8 m^3$ ; $\times m^2 = 8 m^5$ .
$5 x^3$ .
$10 a^7$ .
$27 x^6$ .
$a^3 b^4$ .
$x^4$ . Method: top $x^{12} \cdot x^2 = x^{14}$ ; $x^{14} \div x^{10} = x^4$ .

Reasoning

Explain and spot the mistake

Wrong. $2^3 \times 2^3 = 2^{3 + 3} = 2^6 = 64$ . Sam added $3 \times 3$ or used the wrong law.
Dividing anything non-zero by itself is $1$ . The quotient rule says $\dfrac{a^n}{a^n} = a^{n-n} = a^0$ . So $a^0 = 1$ .
Wrong. $(a^2)^3 = a^{2 \times 3} = a^6$ (power of a power multiplies, not adds, the exponents).
Not formally Year 8 (appears in Year 9). Using the quotient rule consistently: $\dfrac{x^3}{x^5} = x^{-2}$ , which means $\dfrac{1}{x^2}$ .

Problem solving

Applications

$2^n$ cells; after $8$ hours: $2^8 = 256$ cells.
Volume becomes $(2s)^3 = 8 s^3$ , so $8$ times larger.
$2^{10}$ times bigger. ( $2^{20} \div 2^{10} = 2^{10} = 1024$ .)
$100 n^2$ cm², or $10^2 n^2$ .

Challenge - answers

Reasoning

Harder reasoning

$4 x^5 y^3$ . Method: top $= 4 x^6 y^2 \times x^3 y^6 = 4 x^9 y^8$ ; divide by $x^4 y^5$ → $4 x^5 y^3$ .
$a^{m+n} = a^m \times a^n = 8 \times 4 = 32$ .
$P = 5000 \times 2^6 = 5000 \times 64 = 320\,000$ .
$2^{11}$ . Method: $8 = 2^3$ , $4 = 2^2$ , $32 = 2^5$ . Expression becomes $\dfrac{2^{15} \times 2^6}{2^{10}} = \dfrac{2^{21}}{2^{10}} = 2^{11}$ .

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