Exponentiation facts for kids

Kids Encyclopedia Facts

Exponentiation is a special math operation that helps us multiply a number by itself many times. Think of it like a super-fast way to do repeated multiplication, just as multiplication is a fast way to do repeated addition!

When you see a number like $x^y$ , it means you multiply the number $x$ by itself $y$ times. We read this as " $x$ raised to the power of $y$ " or " $x$ to the $y$ th power."

In $x^y$ :

The number $x$ is called the base. It's the number being multiplied.
The number $y$ is called the exponent (or power). It tells you how many times to multiply the base by itself.

For example, in $2^4$ , the base is 2 and the exponent is 4. To figure out $2^4$ , you multiply 2 by itself 4 times: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^4 = 2 \cdot 2 \cdot 2 \cdot 2 And the answer is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \cdot 2 \cdot 2 \cdot 2 = 16 . So, "2 raised to the power of 4 equals 16."

Here are some more examples:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 5^3 = 5 \cdot 5 \cdot 5 = 125
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 = x \cdot x
$1^x = 1$ (This means 1 multiplied by itself any number of times is always 1!)

Special Exponents
Rules of Exponents
Commutativity
Inverse Operations
Related pages
See also

Special Exponents

Some exponents have special names or meanings.

The Square (Exponent 2)

If the exponent is 2, we call it a square. For example, $x^2$ is the square of $x$ . This name comes from geometry! The area of a square shape is found by multiplying its side length by itself. So, if a square has a side length of a, its area is $a^2$ .

The Cube (Exponent 3)

If the exponent is 3, we call it a cube. For example, $x^3$ is the cube of $x$ . This also comes from geometry! The volume of a cube shape is found by multiplying its side length by itself three times. So, if a cube has a side length of a, its volume is $a^3$ .

Negative Exponents

When the exponent is a negative number, it means you take the reciprocal of the base. The reciprocal of a number is 1 divided by that number.

If the exponent is -1, it's simply the reciprocal:

$x^{-1} = \frac{1}{x}$

If the exponent is any negative integer (whole number), you take the reciprocal and then raise it to the positive version of that exponent:

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

Fractional Exponents (Roots)

Sometimes, the exponent can be a fraction!

If the exponent is $\tfrac{1}{2}$ , it means you find the square root of the base:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^{\frac{1}{2}} = \sqrt{x} For example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 4^{\frac{1}{2}} = \sqrt{4} = 2 (because Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \cdot 2 = 4 ).

If the exponent is $\tfrac{1}{n}$ , it means you find the nth root:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a^{\frac{1}{n}} = \sqrt[n]{a}

If the exponent is a rational number (a fraction) like $\tfrac{p}{q}$ , it means you take the qth root of the base, and then raise that result to the power of p:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a^{\frac{p}{q}} = \sqrt[q]{a^p}

Rules of Exponents

There are some helpful rules that make working with exponents easier:

Multiplying bases with the same exponent:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \left(a \cdot b\right)^n = a^n \cdot b^n Example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (2 \cdot 3)^2 = 6^2 = 36 . Also, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^2 \cdot 3^2 = 4 \cdot 9 = 36 .

Dividing bases with the same exponent:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} (as long as $b$ is not 0) Example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \left(\frac{6}{3}\right)^2 = 2^2 = 4 . Also, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{6^2}{3^2} = \frac{36}{9} = 4 .

Multiplying powers with the same base: Add the exponents.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a^r \cdot a^s = a^{r+s} Example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^3 \cdot 2^2 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2) = 2^5 = 32 . Using the rule: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^{3+2} = 2^5 = 32 .

Dividing powers with the same base: Subtract the exponents.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{a^r}{a^s} = a^{r-s} (as long as $a$ is not 0) Example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{2^5}{2^2} = \frac{32}{4} = 8 . Using the rule: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^{5-2} = 2^3 = 8 .

Negative exponent rule:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a^{-n} = \frac{1}{a^n} (as long as $a$ is not 0) Example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^{-2} = \frac{1}{2^2} = \frac{1}{4} .

Power of a power: Multiply the exponents.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \left(a^r\right)^s = a^{r \cdot s} Example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (2^2)^3 = (4)^3 = 4 \cdot 4 \cdot 4 = 64 . Using the rule: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^{2 \cdot 3} = 2^6 = 64 .

Zero exponent: Any non-zero number raised to the power of 0 is 1.

$a^0 = 1$ Example: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 5^0 = 1 .

Commutativity

In math, an operation is "commutative" if the order of the numbers doesn't change the answer.

Addition is commutative: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 + 3 is the same as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3 + 2 (both are 5).
Multiplication is commutative: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \cdot 3 is the same as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3 \cdot 2 (both are 6).

But exponentiation is not commutative! The order of the base and exponent matters a lot. For example:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^3 = 2 \cdot 2 \cdot 2 = 8
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3^2 = 3 \cdot 3 = 9

As you can see, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 8 \neq 9 , so $2^3$ is not the same as $3^2$ .

Inverse Operations

Just like addition has subtraction as its opposite (inverse) operation, and multiplication has division, exponentiation also has inverse operations. But because exponentiation is not commutative, it actually has two different inverse operations!

Let's look at examples:

If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x + 2 = 3 , you use subtraction to find Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = 3 - 2 , so $x = 1$ .

If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 + x = 3 , you still use subtraction to find Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = 3 - 2 , so $x = 1$ . (This works because addition is commutative).

If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x \cdot 2 = 3 , you use division to find Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = \frac{3}{2} .

If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \cdot x = 3 , you still use division to find Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = \frac{3}{2} . (This works because multiplication is commutative).

Now for exponentiation:

* If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^2 = 3 (you're looking for the base), you use the root operation to find $x$ . You get Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = \sqrt{3} . * However, if you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^x = 3 (you're looking for the exponent), you cannot use a root. Instead, you use an operation called the logarithm. You would write this as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = \log_2(3) . The logarithm tells you "what power do I need to raise the base (2) to, to get the number (3)?"