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# Exponents

You're probably familiar with the four basic operations (addition, subtraction, multiplication, and division), but did you know there's a whole lot more out there? We'll be going over a new one today. It's called exponentiation and is noted by a superscript such as ${x}^{2}$ or ${e}^{x}$ . If you're using a calculator or computer, the ^ symbol generally means exponent.

When working with exponents, it's important to understand exactly what you're dealing with. For example, you would read ${6}^{2}$ as "6 to the second power" because 6 is the base and 2 is the exponent. You can also call it "6 squared" because it represents the area of a square with sides measuring 6 square units (see square roots for more info). Exponents tell you to multiply the base by itself the exponent number of times, so:

${6}^{2}=6×6=36$

Multiplying 6 and 2 when you see ${6}^{2}$ is a common mistake to make, so be careful to avoid it!

## Working with exponents

Naturally, things become a little more complex when you have an exponent greater than 2. For instance, consider ${3}^{4}$ . The base is 3, which we multiply by itself four times since the exponent is 4. That gives us:

${3}^{4}=3×3×3×3=81$

You may also be asked to determine the greater number between two exponents. For example, is ${4}^{5}$ or ${5}^{4}$ greater? The only way to figure it out is to work out both numbers:

$4×4×4×4×4=1024$

$5×5×5×5=625$

1024 is greater than 625, so ${4}^{5}$ is greater than ${5}^{4}$ .

Two of the most common uses of exponents in mathematics are the powers of 2 and square numbers. We recommend using the chart below to help you memorize what they are:

 Powers of 2 Square Numbers ${2}^{1}=2$ ${1}^{2}=1$ ${2}^{2}=4$ ${2}^{2}=4$ ${2}^{3}=8$ ${3}^{2}=9$ ${2}^{4}=16$ ${4}^{2}=16$ ${2}^{5}=32$ ${5}^{2}=25$ ${2}^{6}=64$ ${6}^{2}=36$ ${2}^{7}=128$ ${7}^{2}=49$ ${2}^{8}=256$ ${8}^{2}=64$ ${2}^{9}=512$ ${9}^{2}=81$ ${2}^{10}=1024$ ${10}^{2}=100$

## Uses for exponents

If you're working with more complicated algebraic expressions rather than the individual exponents above, the properties of exponents could make your work a little easier. Similarly, exponent tables and patterns can help simplify complicated problems.

Scientific notation is also an important use for exponents, especially when you're working with really big or small numbers. Scientific notation is rooted in the bases of 10 and includes examples such as $4.56×{10}^{3}=4560$ and $3.802×{10}^{-5}=0.00003802$ .

## Practice questions on exponents

a. What is ${6}^{7}$ ?

$279936$

b. Which is greater: ${5}^{6}$ or ${6}^{5}$ ?

${5}^{6}$

c. Expanding on the chart above for patterns, what is ${2}^{11}$ ?

$2048$

d. What about ${11}^{2}$ ?

$121$

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