Multiplying and dividing powers

Work­ing out $5^3 \times 5^7$ by mul­ti­ply­ing every­thing out would take ages. But there is no need. Pow­ers fol­low a few sim­ple rules called the laws of expo­nents. This les­son cov­ers the first two: what hap­pens when you mul­ti­ply pow­ers of the same base, and what hap­pens when you divide them.

Law 1: mul­ti­ply­ing pow­ers with the same base#

Look at $2^2 \times 2^3$ writ­ten out in full.

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

We had two 2s and then three more 2s, which makes $2 + 3 = 5$ twos. So $2^2 \times 2^3 = 2^{2+3} = 2^5$.

Check with num­bers: $2^2 \times 2^3 = 4 \times 8 = 32$, and $2^5 = 32$.

Law 1. $a^m \times a^n = a^{m+n}$. When you mul­ti­ply pow­ers of the same base, keep the base and add the expo­nents.

Exam­ple 1#

$$\begin{aligned} 5^3 \times 5^7 &= 5^{3+7} = 5^{10} \\ 2^3 \times 2^2 &= 2^{3+2} = 2^5 \\ 5^3 \times 5^3 &= 5^{3+3} = 5^6 \\ 7^8 \times 7^8 &= 7^{8+8} = 7^{16} \\ 2^{13} \times 2^{25} &= 2^{13+25} = 2^{38} \end{aligned}$$

Exam­ple 2: a frac­tion as the base#

The law works for any base, includ­ing a frac­tion.

$$\displaystyle \left(\frac{1}{2}\right)^8 \times \left(\frac{1}{2}\right)^3 = \left(\frac{1}{2}\right)^{8+3} = \left(\frac{1}{2}\right)^{11}$$

Exam­ple 3: three pow­ers#

With more than two pow­ers, add all the expo­nents.

$$10^2 \times 10^3 \times 10^5 = 10^{2+3+5} = 10^{10}$$

$$7^{12} \times 7^{16} \times 7^{45} = 7^{12+16+45} = 7^{73}$$

Only for the same base#

The law needs the same base. $2^3 \times 3^2$ can­not be joined into one power, because the bases $2$ and $3$ are dif­fer­ent. You must work it out: $8 \times 9 = 72$.

Law 2: divid­ing pow­ers with the same base#

Now look at $\displaystyle \frac{2^5}{2^2}$ writ­ten out in full.

$$\displaystyle \frac{2^5}{2^2} = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2} = 2 \times 2 \times 2 = 2^3$$

Two 2s on the bot­tom can­cel two of the five 2s on top. That leaves $5 - 2 = 3$ twos. So $\displaystyle \frac{2^5}{2^2} = 2^{5-2} = 2^3$.

Law 2. $\displaystyle \frac{a^m}{a^n} = a^{m-n}$, where $a \neq 0$. When you divide pow­ers of the same base, keep the base and take away the expo­nents.

The base can­not be $0$, because we can never divide by $0$.

Exam­ple 4#

$$\displaystyle \begin{aligned} \frac{8^9}{8^7} &= 8^{9-7} = 8^2 = 64 \\ \frac{3^4}{3^1} &= 3^{4-1} = 3^3 \\ \frac{100^5}{100^2} &= 100^{5-2} = 100^3 \\ \frac{15^7}{15^4} &= 15^{7-4} = 15^3 \end{aligned}$$

Exam­ple 5: a base with no expo­nent shown#

Find $\displaystyle \frac{10^8}{10}$.

A num­ber with no expo­nent shown has expo­nent $1$: $10 = 10^1$.

$$\displaystyle \frac{10^8}{10} = \frac{10^8}{10^1} = 10^{8-1} = 10^7$$

Exam­ple 6: big expo­nents#

$$\displaystyle \frac{5^{25}}{5^{17}} = 5^{25-17} = 5^8 \qquad \frac{5^{2013}}{5^{178}} = 5^{2013-178} = 5^{1835}$$

You never need to work out these huge num­bers. The law does the job with one sub­trac­tion.

Using both laws together#

Exam­ple 7#

Find $\displaystyle \frac{5^3 \times 5^{18}}{5^{12}}$.

First the top, by Law 1. Then the divi­sion, by Law 2.

$$\displaystyle \frac{5^3 \times 5^{18}}{5^{12}} = \frac{5^{21}}{5^{12}} = 5^{21-12} = 5^9$$

Exam­ple 8#

Find $\displaystyle \frac{2^3 \times 2^4}{2^5}$ as a num­ber.

$$\displaystyle \frac{2^3 \times 2^4}{2^5} = \frac{2^7}{2^5} = 2^2 = 4$$

Com­mon mis­takes#

• Mul­ti­ply­ing the expo­nents instead of adding them. $2^5 \times 2^7 = 2^{12}$, not $2^{35}$.
• Mul­ti­ply­ing the bases as well. $2^5 \times 2^7$ is $2^{12}$, not $4^{12}$. The base stays the same.
• Using the law with dif­fer­ent bases. $2^3 \times 3^2$ is not $6^5$.
• For­get­ting that $10$ means $10^1$.

Prac­tice#

1. Sim­plify $2^5 \times 2^7$.
2. Sim­plify $3^2 \times 3^4 \times 3^8$.
3. Sim­plify $\displaystyle \frac{6^{15}}{6^{10}}$.
4. Sim­plify $\displaystyle \frac{5^{13}}{5^8}$.
5. Sim­plify $\displaystyle \left(\frac{2}{3}\right)^4 \times \left(\frac{2}{3}\right)^5$.
6. Sim­plify $\displaystyle \frac{10^6}{10}$.
7. Sim­plify $x^4 \times x^9$.
8. Find the value of $\displaystyle \frac{7^{20}}{7^{18}}$.
9. Sim­plify $\displaystyle \frac{9^{49}}{9^{37}}$.
10. Sim­plify $\displaystyle \frac{3^5 \times 3^6}{3^9}$ and find its value.

Answers#