Simplifying with the laws of exponents

You now know all eleven laws of expo­nents. Real ques­tions mix them: a power of a power inside a divi­sion, neg­a­tive expo­nents inside a sum, frac­tions raised to pow­ers. This les­son first lists the laws in one place, shows how each one can be used back­wards, and then works through mixed ques­tions step by step.

The eleven laws#

The laws used back­wards#

Every law can be read from right to left. This is often the key to a ques­tion.

$$\displaystyle \begin{aligned} 2^{x+y} &= 2^x \times 2^y & &\text{Law 1 backwards} \\ 3^{x-2} &= \frac{3^x}{3^2} = \frac{3^x}{9} & &\text{Law 2 backwards} \\ 2^{2x} &= (2^2)^x = 4^x & &\text{Law 3 backwards} \\ (2x)^2 &= 2^2 \times x^2 = 4x^2 & &\text{Law 4 backwards} \\ \left(\frac{2}{x}\right)^2 &= \frac{2^2}{x^2} = \frac{4}{x^2} & &\text{Law 5 backwards} \end{aligned}$$

And for the other laws:

• $1 = 2^0$ (Law 6), and $\displaystyle \frac{1}{2} = 2^{-1}$ (Law 7).
• $\displaystyle \left(\frac{2}{3}\right)^2 = \left(\frac{3}{2}\right)^{-2}$ (Law 8).
• $-1 = (-1)^5$ and $1 = (-1)^{10}$ (Law 9).
• $-2^3 = (-2)^3$ and $2^4 = (-2)^4$ (Law 10).

Mixed ques­tions#

Work from the inside out: brack­ets first, then pow­ers of pow­ers, then mul­ti­ply­ing and divid­ing.

Exam­ple 1#

Find $(2^{20} \div 2^{15}) \times 2^3$.

$$(2^{20} \div 2^{15}) \times 2^3 = 2^5 \times 2^3 = 2^8 = 256$$

Exam­ple 2#

Find $(3^{-7} \div 3^{-10}) \times 3^{-5}$.

$$\displaystyle \begin{aligned} (3^{-7} \div 3^{-10}) \times 3^{-5} &= 3^{-7+10} \times 3^{-5} \\ &= 3^3 \times 3^{-5} \\ &= 3^{-2} = \frac{1}{9} \end{aligned}$$

Exam­ple 3#

Find $\displaystyle \frac{(2^2)^3 \times 2^5}{2^7}$.

$$\displaystyle \frac{(2^2)^3 \times 2^5}{2^7} = \frac{2^6 \times 2^5}{2^7} = \frac{2^{11}}{2^7} = 2^4 = 16$$

Exam­ple 4#

Find $\displaystyle \frac{12^4 \times 9^4 \times 4^4}{6^3 \times 8^3 \times 3^3}$.

Use Law 4 on the top and on the bot­tom.

$$\displaystyle \frac{12^4 \times 9^4 \times 4^4}{6^3 \times 8^3 \times 3^3} = \frac{(12 \times 9 \times 4)^4}{(6 \times 8 \times 3)^3} = \frac{432^4}{144^3}$$

Now notice that $432 = 3 \times 144$. So $432^4 = 3^4 \times 144^4$, and

$$\displaystyle \frac{432^4}{144^3} = \frac{3^4 \times 144^4}{144^3} = 3^4 \times 144 = 81 \times 144 = 11664$$

Exam­ple 5#

Find $\displaystyle 3^4 \times \left(\frac{5}{3}\right)^4$.

Both have the power $4$, so use Law 4.

$$\displaystyle 3^4 \times \left(\frac{5}{3}\right)^4 = \left(3 \times \frac{5}{3}\right)^4 = 5^4 = 625$$

Exam­ple 6#

Find $(3^0 + 4^{-1}) \times 2^2$.

$$\displaystyle \begin{aligned} (3^0 + 4^{-1}) \times 2^2 &= \left(1 + \frac{1}{4}\right) \times 4 \\ &= \frac{5}{4} \times 4 \\ &= 5 \end{aligned}$$

Exam­ple 7#

Find $(2^{-1} \times 4^{-1}) \div 2^{-2}$.

$$\displaystyle \begin{aligned} (2^{-1} \times 4^{-1}) \div 2^{-2} &= \left(\frac{1}{2} \times \frac{1}{4}\right) \div \frac{1}{4} \\ &= \frac{1}{8} \times \frac{4}{1} = \frac{1}{2} \end{aligned}$$

Exam­ple 8#

Find $\displaystyle \left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2}$.

By Law 8, each frac­tion turns upside down.

$$2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29$$

Exam­ple 9#

Find $(4^{-1} - 5^{-1}) \div 3^{-1}$.

$$\displaystyle \begin{aligned} (4^{-1} - 5^{-1}) \div 3^{-1} &= \left(\frac{1}{4} - \frac{1}{5}\right) \div \frac{1}{3} \\ &= \frac{5 - 4}{20} \div \frac{1}{3} \\ &= \frac{1}{20} \times 3 = \frac{3}{20} \end{aligned}$$

Exam­ple 10#

Sim­plify $(2x^2)^3 \times (5x)^2$.

$$\begin{aligned} (2x^2)^3 \times (5x)^2 &= 2^3 (x^2)^3 \times 5^2 x^2 \\ &= 8x^6 \times 25x^2 \\ &= 200x^{6+2} = 200x^8 \end{aligned}$$

Exam­ple 11#

Find $\displaystyle 4^3 \times \left(\frac{3}{2}\right)^6$.

$4^3 = 64$ and $2^6 = 64$, so they can­cel.

$$\displaystyle 4^3 \times \left(\frac{3}{2}\right)^6 = 64 \times \frac{3^6}{2^6} = 64 \times \frac{729}{64} = 729$$

Exam­ple 12#

Find $\displaystyle \left(\frac{2}{5}\right)^{-2} \times \left(\frac{7}{2}\right)^{-5} \times \left(\frac{5}{7}\right)^{-3}$.

Turn each frac­tion upside down (Law 8), then split each power (Law 5).

$$\displaystyle \begin{aligned} &\left(\frac{5}{2}\right)^2 \times \left(\frac{2}{7}\right)^5 \times \left(\frac{7}{5}\right)^3 \\ &= \frac{5^2}{2^2} \times \frac{2^5}{7^5} \times \frac{7^3}{5^3} \\ &= \frac{2^5}{2^2} \times \frac{5^2}{5^3} \times \frac{7^3}{7^5} \\ &= 2^3 \times 5^{-1} \times 7^{-2} \\ &= \frac{8}{5 \times 49} = \frac{8}{245} \end{aligned}$$

Exam­ple 13: frac­tions in the expo­nent#

The laws also work when the expo­nents are frac­tions, as long as the base is pos­i­tive. You will learn what a frac­tional expo­nent means in a later chap­ter. For now, just add and take away.

$$\displaystyle 5^{\frac{3}{2}} \times 5^{\frac{5}{2}} = 5^{\frac{3}{2} + \frac{5}{2}} = 5^{\frac{8}{2}} = 5^4 = 625$$

$$\displaystyle \frac{5^{\frac{11}{2}} \times 5^{-\frac{3}{2}}}{5^5} = \frac{5^{\frac{8}{2}}}{5^5} = \frac{5^4}{5^5} = 5^{-1} = \frac{1}{5}$$

Exam­ple 14#

Find $(2^5 \div 2^8)^5 \times 2^{-5}$.

$$\displaystyle \begin{aligned} (2^5 \div 2^8)^5 \times 2^{-5} &= (2^{-3})^5 \times 2^{-5} \\ &= 2^{-15} \times 2^{-5} \\ &= 2^{-20} = \frac{1}{2^{20}} \end{aligned}$$

Exam­ple 15#

Find $x$ if $\displaystyle \left(-\frac{1}{2}\right)^{-19} \div \left(-\frac{1}{2}\right)^8 = \left(-\frac{1}{2}\right)^{-2x+1}$.

The base is the same on both sides, and it is not $-1$, $0$ or $1$.

$$\displaystyle \begin{aligned} \left(-\frac{1}{2}\right)^{-19-8} &= \left(-\frac{1}{2}\right)^{-2x+1} \\ \Rightarrow -27 &= -2x + 1 \\ \Rightarrow 2x &= 1 + 27 = 28 \\ \Rightarrow x &= 14 \end{aligned}$$

Com­mon mis­takes#

• Doing the steps in the wrong order. Brack­ets and pow­ers of pow­ers come first.
• Adding unlike pow­ers. $2^2 + 3^2 + 4^2$ is $29$, not $9^2$.
• For­get­ting that $a^0 = 1$ inside a sum: $\displaystyle 3^0 + 4^{-1} = 1 + \frac{1}{4}$.
• Using frac­tional expo­nents with a neg­a­tive base. A power like $\displaystyle (-1)^{\frac{1}{2}}$ is not a real num­ber, so the laws can­not be used there.

Prac­tice#

1. Sim­plify $3^{x+2}$ as a num­ber times $3^x$.
2. Find $\displaystyle \frac{(3^2)^3 \times 3^4}{3^8}$.
3. Find $5^{-2} \times 5^4$.
4. Find $2^3 \times 2^{-5} \times 2^4$.
5. Find $3^{-1} + 4^{-1}$ as a frac­tion.
6. Find $(2^{-1} + 3^{-1})^{-1}$.
7. Find $\displaystyle \left(\frac{2}{3}\right)^2 \times \left(\frac{2}{3}\right)^{-4}$.
8. Find $\displaystyle \frac{2^6 \times 3^2}{6^2}$.
9. Find $\displaystyle \left(\frac{1}{3}\right)^{-2} - \left(\frac{1}{2}\right)^{-3}$.
10. Find $\displaystyle \frac{(-2)^{-19}}{(-2)^{-17}}$.

Answers#