Exponents

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Exponents

An exponent has two parts to it, an exponent and a base. The small number to the right and above the number is an exponent (when typing, this is a superscript). The large number underneath is called a base number. The following example shows 3 as the base and 2 as the exponent. Understanding what is the base number and exponent number is the first step.

We learned multiplication by breaking down the equation and adding the multiplication together.

3*5 = 5 + 5 + 5 = 15

This can also be applied to exponents. An exponent multiplies its base number to itself the quantity shown by its exponent. See the following examples to illustrate.

\begin{array}{l} {2^4} = 2*2*2*2 = 16\\ {3^2} = 3*3 = 9\\ {5^3} = 5*5*5 = 125 \end{array}

Writing an equation in exponential form can be converted when the same number is multiplied. The following equation has 7 multiplied by itself 8 times, which equals 7 to the 8th power. This also illustrates the need for exponents to simplify equations, as this expression is shortened from 8 numbers to 2 numbers.

7*7*7*7*7*7*7*7 = {7^8}

Positive and Negative Base Numbers

Positive 1 and negative 1 to different powers can have different results. Positive 1 to different powers always equals 1. Negative 1 to different powers will have alternating results.  A negative number to an even power will be positive because the negatives cancel, whereas an odd number will result in a negative 1.

\begin{array}{l} {1^0} = 1\\ {1^1} = 1\\ {1^2} = 1*1 = 1\\ {1^5} = 1*1*1*1*1 = 1 \end{array}
\begin{array}{l} - {1^0} = 1\\ - {1^1} = - 1\\ - {1^2} = - 1* - 1 = 1\\ - {1^3} = - 1* - 1* - 1 = - 1\\ - {1^4} = - 1* - 1* - 1* - 1 = 1 \end{array}

Squared Numbers

Often, we multiply a number by itself once. This can be called the 2nd power, but it is called the square number. The following 4, 9, and 16 square numbers are found by squaring 2, 3, and 4.

\begin{array}{l} 4 = {2^2}\\ 9 = {3^2}\\ 16 = {4^2} \end{array}

The First Power

Base numbers can also be to the 0 or the first power. When any non-zero number is raised to the 0 power, the answer is always 1. When any non-zero number is raised to the first power, the answer is always the base number.

\begin{array}{c} {2^0} = 1\\ {5^0} = 1\\ {2^1} = 2\\ {5^1} = 5 \end{array}

Powers of zero always equal zero, as the multiplication multiplies zero by the exponential value.

\begin{array}{c} {0^1} = 0\\ {0^2} = 0*0 = 0\\ {0^3} = 0*0*0 = 0 \end{array}

Zero to the Zero Power

Zero to the zero power differs depending on whom you ask. Most algebraic applications agree that the value of zero to the zero power is 1. However, in mathematical analysis or computer programming zero to the zero power is undefined.

\begin{array}{c} {0^0} = 1\\ {0^0} = undefined \end{array}

Exponent with Decimal Base

Multiplying decimals follows the same pattern as the other exponential rules.

\begin{array}{c} {0.3^2} = 0.3*0.3 = 0.09\\ {0.5^3} = 0.5*0.5*0.5 = 0.125\\ {0.1^5} = 0.1*0.1*0.1*0.1*0.1 = 0.00001 \end{array}

To evaluate an expression with exponents as variables, the variable needs to be replaced with the value of the variable and solved.

\begin{array}{l} x = 2\\ {7^x} - {3^x}\\ {7^2} - {3^2}\\ 49 - 9 = 40 \end{array}

Properties of Exponents

There are three different properties of powers or exponents. For all of these to be applicable, the base number must be the same.

Product of Powers Property

The product of powers property is one method to simplify exponents during multiplication. In this instance, the exponents are added.

{a^b}*{a^c} = {a^{(b + c)}}
{3^3}*{3^5} = {3^{(3 + 5)}} = {3^8}

Power of a Power Property

The power of a power property is one method to simplify exponents with an exponent. In this instance, the exponents can be multiplied to simplify.

{({a^b})^c} = {a^{b*c}}
{({3^2})^4} = ({3^2})*({3^2})*({3^2})*({3^2}) = {3^{2 + 2 + 2 + 2}} = {3^{2*4}} = {3^8}

Quotient of Powers Property

The quotient of powers property is one method to simplify exponents during division. In this instance, the exponents are subtracted.

\frac{{{a^b}}}{{{a^c}}} = {a^{b - c}}
\frac{{{3^7}}}{{{3^4}}} = {3^{(7 - 4)}} = {3^3}

Frequently Asked Questions

What does a negative exponent mean?

A negative exponent is an inverse (opposite) of multiplying the base number, it means dividing the base number.

{3^{ - 1}} = \frac{1}{3}
{4^{ - 3}} = \frac{1}{{(4*4*4)}} = \frac{1}{{{4^3}}}

How to multiply exponents?

See the product of powers property on how to multiply exponents with the same base earlier on this page.

Multiplying exponents with different bases have two different methods. When the base number is different and the exponent is the same, the base numbers can be multiplied.

{a^b}*{c^b} = {(a*c)^b}
{3^2}*{4^2} = {(3*4)^2}

When the base number is different and the exponent is different, there is no simplification that can be completed. The multiplication needs to be fully worked out.

{a^b}*{c^d} = {a^b}*{c^d}
{2^3}*{4^5} = (2*2*2)*(4*4*4*4*4) = (8)*(1024)

How to divide exponents?

See the quotient of powers property on how to divide exponents with the same base earlier on this page.

Dividing exponents with different bases have two different methods. When the base number is different and the exponent is the same, the base numbers can be divided.

\frac{{{a^b}}}{{{c^b}}} = {\left( {\frac{a}{c}} \right)^b}
\frac{{{3^2}}}{{{4^2}}} = {\left( {\frac{3}{4}} \right)^2}

When the base number is different and the exponent is different, there is no simplification that can be completed. The division needs to be fully worked out.

\frac{{{a^b}}}{{{c^d}}} = \frac{{{a^b}}}{{{c^d}}}
\frac{{{3^4}}}{{{5^2}}} = \frac{{(3*3*3*3)}}{{(5*5)}} = \frac{{81}}{{25}}

How to type exponents?

Highlight the exponent, open the document font editor, and change the font to ‘Superscript’.

In a word document, highlight the exponent, and press CTRL, SHIFT, =.

What is 2 squared?

{2^2} = 2*2 = 4

More Information

Check out the Algebra Index for more Algebra topics!

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