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Exponent facts for kids

Kids Encyclopedia Facts

An exponent is a special number that tells you how many times to multiply a number by itself. Think of it as a shortcut for repeated multiplication!

For example, in the number 5^4, the number 5 is called the base, and the small number 4 is the exponent. You can read this as "5 to the power of 4". It means you multiply 5 by itself 4 times: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 5 \cdot 5 \cdot 5 \cdot 5 = 3125 .

In general, if you see x^y, it means you multiply the base number x by itself y times.

When a number is raised to the power of two, like x^2, we often say it's "squared". This is because if you have a square with sides of length x, its area is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x \cdot x , or x^2.

Similarly, when a number is raised to the power of three, like x^3, we say it's "cubed". This is because a cube with sides of length x has a volume of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x \cdot x \cdot x , or x^3.

Exponents are super useful in algebra and other parts of mathematics. They help us write very long multiplication problems in a much shorter way.

Basic Rules of Exponents

There are a few simple rules that help us work with exponents. Let's look at them!

Product Rule: Multiplying Exponents

When you multiply numbers with the same base, you can add their exponents.

The rule is: a^m \cdot a^n = a^{m+n}

  • What it means:* If you have a multiplied by itself m times, and you multiply that by a multiplied by itself n times, you end up with a multiplied by itself a total of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): m+n times.
  • Example:*

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3^2 \cdot 3^4 This means Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3) . If you count all the 3s, there are 6 of them. So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3^2 \cdot 3^4 = 3^6 . Notice that Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 + 4 = 6 , which matches the rule!

Quotient Rule: Dividing Exponents

When you divide numbers with the same base, you can subtract their exponents.

The rule is: \frac{a^m}{a^n}=a^{m-n}

  • What it means:* If you have a multiplied m times on top and a multiplied n times on the bottom, you can cancel out n of the a's from both top and bottom. You are left with a multiplied Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): m-n times.
  • Example:*

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{4^4}{4^2} This means Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{4 \cdot 4 \cdot 4 \cdot 4}{4 \cdot 4} . You can cancel two 4s from the top and two 4s from the bottom. You are left with Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 4 \cdot 4 , which is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 4^2 . Notice that Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 4 - 2 = 2 , which matches the rule!

Sometimes, when you subtract the exponents, you might get a negative number. We'll learn about that next!

Zero Rule: Exponent of Zero

Any number (except zero) raised to the power of zero is always 1.

The rule is: b^0 = 1 (where b is not zero)

  • Why it works:*

Let's use the Quotient Rule. We know that Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{b^m}{b^m} = b^{m-m} = b^0 . But we also know that any number divided by itself is 1 (as long as it's not zero). So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{b^m}{b^m} = 1 . This means b^0 must be equal to 1!

Negative Exponents

A negative exponent means you take the reciprocal of the number with a positive exponent. The reciprocal of a number is 1 divided by that number.

The rule is: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^{-a}=\frac{1}{x^{a}}

  • What it means:* Instead of multiplying, a negative exponent tells you to divide. For example, x^{-1} means 1/x.
  • Example:*

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 5^{-1} = \frac{1}{5^1} = \frac{1}{5} Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^{-3} = \frac{1}{2^3} = \frac{1}{2 \cdot 2 \cdot 2} = \frac{1}{8}

  • Why it works:*

Let's think about the Product Rule again: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b^x \cdot b^{-x} = b^{x+(-x)} = b^0 . We already learned that b^0 = 1. So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b^x \cdot b^{-x} = 1 . To find out what Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b^{-x} is, we can divide both sides by b^x: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{b^x \cdot b^{-x}}{b^x} = \frac{1}{b^x} This leaves us with Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b^{-x} = \frac{1}{b^x} .

If you have a number multiplied by a term with a negative exponent, like Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2x^{-3} , it means Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \cdot \frac{1}{x^3} , which is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{2}{x^3} .

Related pages

See also

Kids robot.svg In Spanish: Potenciación para niños

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