Power of a power, product and quotient

The first two laws dealt with pow­ers of the same base side by side. This les­son adds three more laws. The third law tells you what to do with a power of a power, such as $(2^3)^2$. The fourth and fifth laws deal with dif­fer­ent bases raised to the same power, such as $2^3 \times 5^3$.

Law 3: a power of a power#

Look at $(2^3)^2$. The outer $2$ says: write $2^3$ twice and mul­ti­ply.

$$\begin{aligned} (2^3)^2 &= 2^3 \times 2^3 \\ &= (2 \times 2 \times 2) \times (2 \times 2 \times 2) \\ &= 2^6 \end{aligned}$$

Two groups of three 2s make $3 \times 2 = 6$ twos. So $(2^3)^2 = 2^{3 \times 2} = 2^6$.

Law 3. $(a^m)^n = a^{m \times n}$. Read it as "a power m, whole power n". For a power of a power, mul­ti­ply the expo­nents.

Exam­ple 1#

$$\begin{aligned} (9^2)^3 &= 9^{2 \times 3} = 9^6 \\ (8^{10})^{10} &= 8^{10 \times 10} = 8^{100} \\ (3^2)^5 &= 3^{2 \times 5} = 3^{10} \\ (5^3)^9 &= 5^{3 \times 9} = 5^{27} \\ (24^3)^{123} &= 24^{3 \times 123} = 24^{369} \end{aligned}$$

Brack­ets mat­ter#

$(2^3)^2$ and $2^{3^2}$ are not the same. $(2^3)^2 = 2^6 = 64$. But $2^{3^2}$ means $2$ to the power $3^2 = 9$, which is $2^9 = 512$.

Law 4: the same power of dif­fer­ent bases (mul­ti­ply­ing)#

Look at $2^3 \times 5^3$.

$$\begin{aligned} 2^3 \times 5^3 &= (2 \times 2 \times 2) \times (5 \times 5 \times 5) \\ &= (2 \times 5) \times (2 \times 5) \times (2 \times 5) \\ &= (2 \times 5)^3 = 10^3 \end{aligned}$$

We paired each $2$ with a $5$. There are three pairs, so we get $(2 \times 5)^3$.

Law 4. $a^n \times b^n = (a \times b)^n$, that is, $a^n b^n = (ab)^n$. When the pow­ers are the same, mul­ti­ply the bases and keep the power.

Exam­ple 2#

$$\begin{aligned} 2^2 \times 5^2 &= (2 \times 5)^2 = 10^2 \\ 5^9 \times 8^9 &= (5 \times 8)^9 = 40^9 \\ 9^5 \times 6^5 &= (9 \times 6)^5 = 54^5 \\ 4^9 \times 5^9 &= (4 \times 5)^9 = 20^9 \end{aligned}$$

Exam­ple 3: a clever use#

Find $25^3 \times 4^3$.

$$25^3 \times 4^3 = (25 \times 4)^3 = 100^3 = 10{,}00{,}000$$

Much eas­ier than work­ing out $25^3$ and $4^3$ first.

Law 4 the other way round#

Read the law from right to left: $(ab)^n = a^n \times b^n$. This lets you open a bracket with let­ters inside.

$$\begin{aligned} (2x)^2 &= 2^2 \times x^2 = 4x^2 \\ (3x)^3 &= 3^3 \times x^3 = 27x^3 \\ (ab)^4 &= a^4 b^4 \end{aligned}$$

Exam­ple 4#

Sim­plify $(5y^3)^2$.

Use Law 4 to split the bracket, then Law 3 on $(y^3)^2$.

$$(5y^3)^2 = 5^2 \times (y^3)^2 = 25y^6$$

Law 5: the same power of dif­fer­ent bases (divid­ing)#

Look at $\displaystyle \frac{5^2}{3^2}$.

$$\displaystyle \frac{5^2}{3^2} = \frac{5 \times 5}{3 \times 3} = \frac{5}{3} \times \frac{5}{3} = \left(\frac{5}{3}\right)^2$$

Law 5. $\displaystyle \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$, where $b \neq 0$. When the pow­ers are the same, divide the bases and keep the power.

Exam­ple 5#

$$\displaystyle \frac{2^2}{3^2} = \left(\frac{2}{3}\right)^2 \qquad \frac{5^7}{8^7} = \left(\frac{5}{8}\right)^7$$

$$\displaystyle \frac{10^3}{5^3} = \left(\frac{10}{5}\right)^3 = 2^3 = 8$$

$$\displaystyle \frac{6^2}{4^2} = \left(\frac{6}{4}\right)^2 = \left(\frac{3}{2}\right)^2$$

Law 5 the other way round#

$\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. To raise a frac­tion to a power, raise the top and the bot­tom.

$$\displaystyle \left(\frac{5}{3}\right)^2 = \frac{5^2}{3^2} = \frac{25}{9} \qquad \left(\frac{3}{5}\right)^4 = \frac{3^4}{5^4} = \frac{81}{625}$$

$$\displaystyle \left(\frac{x^2}{2}\right)^2 = \frac{(x^2)^2}{2^2} = \frac{x^4}{4}$$

Using sev­eral laws together#

Exam­ple 6#

Sim­plify $\displaystyle \frac{(2^5)^2 \times 2^7}{2^4}$.

$$\displaystyle \frac{(2^5)^2 \times 2^7}{2^4} = \frac{2^{10} \times 2^7}{2^4} = \frac{2^{17}}{2^4} = 2^{13}$$

Law 3 first, then Law 1, then Law 2.

Exam­ple 7#

Sim­plify $(x y^3)^2 \times (x^3 y)^3$.

$$\begin{aligned} (x y^3)^2 \times (x^3 y)^3 &= x^2 (y^3)^2 \times (x^3)^3 y^3 \\ &= x^2 y^6 \times x^9 y^3 \\ &= x^{2+9} y^{6+3} \\ &= x^{11} y^9 \end{aligned}$$

Com­mon mis­takes#

• Adding expo­nents for a power of a power. $(5^3)^9 = 5^{27}$, not $5^{12}$.
• Using Law 4 when the pow­ers are dif­fer­ent. $2^3 \times 5^2$ is not $10^5$ or $10^3$.
• Rais­ing only part of a bracket. $(2x)^2 = 4x^2$, not $2x^2$.
• Rais­ing only the top of a frac­tion. $\displaystyle \left(\frac{5}{3}\right)^2 = \frac{25}{9}$, not $\displaystyle \frac{25}{3}$.

Prac­tice#

1. Sim­plify $(3^4)^3$.
2. Sim­plify $(7^3)^3$.
3. Write $(10^2)^3$ as a power of $10$ and as a num­ber.
4. Sim­plify $3^4 \times 5^4$.
5. Sim­plify $4^6 \times 25^6$ as a power of $100$.
6. Sim­plify $(2a)^4$.
7. Sim­plify $(5x^3)^2$.
8. Find the value of $\displaystyle \left(\frac{5}{2}\right)^2$.
9. Find the value of $\displaystyle \frac{12^3}{4^3}$.
10. Sim­plify $(x^2 y)^3 \times (x y^2)^2$.

Answers#