# Exponents

Exponents (sometimes called powers) are used when you want to multiply a number (or variable) times itself a certain number of times.

Exponents are written with a base (the number being multiplied by itself) which is written like a normal number, and an exponent (the number that tells you how many times to multiply the base times itself) which is written in a small font, to the upper right of the base.

$\text{Base}\rightarrow \large{17}^{\large{3}\leftarrow\text{exponent}}$

$$\eqalign{&=\large{17^3} &&\text{Means that the base of 17 should be multiplied by itself three times.}\\&=17 \times 17 \times 17\\&=4,913}$$

All positve exponents work the same way. The exponent simply tells you how many times to multiply the base times itself.

$$\eqalign{&=\large{4^6} &&\text{Means that the base of 4 should be multiplied by itself six times.}\\&=4 \times 4 \times 4 \times 4 \times 4 \times 4\\&=4,096\\\text{} \\\text{}\\\text{}\\&=\large{3^2} &&\text{Means that the base of 3 should be multiplied by itself twice}\\&=3 \times 3 \\&=8\\\text{} \\\text{} \\\text{} \\&=\large{12^3} &&\text{Means that the base of 12 should be multiplied by itself three times.}\\&=12 \times 12 \times 12\\&=1,728}$$

Exponents always work the same way, whether you're dealing with

$$\eqalign{\text{Fractions:}\\&=\large({\dfrac{3}{4})^5} &&\text{Means that the base of } \dfrac{3}{4}\text{ should be multiplied by itself 5 times.}\\&=\dfrac{3}{4} \times \dfrac{3}{4} \times \dfrac{3}{4} \times \dfrac{3}{4} \times \dfrac{3}{4}\\&=\dfrac{243}{1024}\\\text{} \\\text{}\\\text{}\\\text{Decimals:}\\&=\large{.03^2} &&\text{Means that the base of .03 should be multiplied by itself}\\&=.03 \\&=.0009\\\text{} \\\text{} \\\text{} \\\text{Variables:}\\&=\large{x^4} &&\text{Means that the base of }x\text{ should be multiplied by itself four times.}\\&=x \times x \times x \times x\\&=x^4\\\text{} \\\text{} \\\text{} \\\text{Or even terms with exponents in them:}\\&=\large{(2x^3)^2} &&\text{Means that the base of }2x^3\text{ should be multiplied by itself.}\\&=(2x^3) \times (2x^3)\\&=4x^6&&\text{For more on this process see Exponents (Multiplying with Exponents)}}$$

If you are using a calculator, see our calculator lessons to see how to calculate exponents on your calculator.

#### Practice Problems:

## Exponents

Write the following numbers/expressions as exponents in simplest form:

- $8\times8$
- $12\times12$
- $2\times2\times2\times2$
- $a\times a\times a$
- $25$
- $81 \times 9$
- $3\times 3\times y\times y\times y$
- $(3y)\times (3y)\times (3y)$
- $3\times y\times y$
- $100 \times x \times x$
- $4$
- $16 \times 64$

#### Answer Key: