Exponent facts for kids
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 , 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 , it means you multiply the base number
by itself
times.
When a number is raised to the power of two, like , we often say it's "squared". This is because if you have a square with sides of length
, its area is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x \cdot x , or
.
Similarly, when a number is raised to the power of three, like , we say it's "cubed". This is because a cube with sides of length
has a volume of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x \cdot x \cdot x , or
.
Exponents are super useful in algebra and other parts of mathematics. They help us write very long multiplication problems in a much shorter way.
Contents
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:
- What it means:* If you have
multiplied by itself
times, and you multiply that by
multiplied by itself
times, you end up with
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:
- What it means:* If you have
multiplied
times on top and
multiplied
times on the bottom, you can cancel out
of the
's from both top and bottom. You are left with
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: (where
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 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,
means
.
- 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 . 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
: 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
In Spanish: Potenciación para niños