## Introduction

- Have you seen any number with another number (positive or negative) on the top right? You might have already; those are called Exponents.
- ${\text{2}}^{3}\text{or}{2}^{-4}\text{}$are some examples of exponents, also called exponent expressions.
- Exponents are nothing but repeated multiplication.
- In ${2}^{3}$, 2 is the base, and 3 is the exponent. In the case of $3{\left(x\right)}^{2}$, 3 is called the coefficient, x is the base, and 2 is the exponent.
- ${2}^{3}$ is read as two to the third power and also as two cubed.
- ${7}^{4}$ is read as seven to the fourth power.
- Exponents can be positive or negative.

## Negative Exponents

**${3}^{-4}$**is an example of negative exponents.- Negative exponents can be solved by finding a reciprocal (dropping that number under 1)
- For example, ${3}^{-4}$ is same as $\frac{1}{{3}^{4}}$ which is same as $\frac{1}{81}$

## Adding & Subtracting Exponents

- If the base and exponents are the same, you only need to add/subtract the coefficient.
- $2{x}^{2}+3{x}^{2}$ can be added, and the result is $5{x}^{2}$.
- $4{x}^{2}-2{x}^{2}$ can be subtracted, and the result is $2{x}^{2}$.
- $3{x}^{2}+4{x}^{4}$ can't be added as the exponents are different.
- $3{x}^{2}-2{y}^{2}$ can't be subtracted as the bases are different.
- $2{\left(3\right)}^{3}+3{\left(2\right)}^{2}$ can be added only by evaluating $2{\left(3\right)}^{3}$ and $3{\left(2\right)}^{2}$ separately and then adding. So it is 18 + 12, which is 30.

## Multiplying Exponents

- To multiply exponents with the same base, keep the base and add the exponent.
- ${a}^{x}\times {a}^{y}$ is the same as ${a}^{\left(x+y\right)}$.
- ${3}^{2}\times {3}^{4}\text{}$is the same as ${3}^{\left(2+4\right)}\text{}$which equates to ${3}^{6}$.
- If the bases are different, simplify each expression and then perform the multiplication.

## Dividing Exponents

- To divide exponents with the same base, keep the base and subtract the exponent.
- ${a}^{x}\xf7{a}^{y}$ is the same as ${a}^{\left(x-y\right)}$.
- ${3}^{4}\xf7{3}^{2}$ is the same as ${3}^{\left(4-2\right)}\text{}$which equates to ${3}^{2}$.
- If the bases are different, simplify each expression and then perform the division.

## Power of Exponents

- To raise an exponent to a power, multiply the exponents.
- ${\left({2}^{3}\right)}^{4}$ is the same as ${\text{2}}^{12}$.
- ${\left({x}^{3}\right)}^{-2}$ is the same as ${x}^{-6}$.

## Exponent Cheat Sheet

**${x}^{0}=1$****${x}^{1}=1$****${x}^{-1}=\frac{1}{x}$****${a}^{m}\times {a}^{n}={a}^{\left(m+n\right)}$****$\frac{{a}^{m}}{{a}^{n}}={a}^{\left(m-n\right)}$**- ${\left({a}^{m}\right)}^{n}={a}^{\left(m\times n\right)}$

## Blunder Area

**${x}^{2}+{x}^{3}$**can't be added directly as the exponents are different.- ${2}^{3}+{3}^{3}$ can't be added directly as the bases are different.
- ${x}^{{3}^{2}}$ is ${x}^{6}$ and not ${x}^{5}$

- Rishi Jethwa
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