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# Exponents

To understand this let us consider the following multiplication

$2 \times 2 \times 2$

We see we have used $2$ three times in this multiplication. This can be written as  $2^3$ the superscript $3$ which is denoted as a small number is called the exponent. The $2$ is called the base.

Now we can say that the exponent is the number of the times the base is used in the multiplication.

For any number $a \neq 0$ we can write an where $a$ is the base and $n$ is the exponent.

$a^n$ is also called as $a$ raised to the power of $n$ or $a$ to the $n^{th}$ power.

We saw that a is not zero. However the $n$ can be zero, negative or positive let us see what happens $n$ = $0$ or negative.

$2^0$ read as $2$ to the power of $0$.

This is a special case as the exponent is zero value is taken as $1$.

So we can say

$a^0$ = $1$

Say if we have $2^{-1}$ this is written as $\frac{1}{2^1}$ that is the $2$ is taken to the denominator and the sign changed.

So for general rule we can have $a^{-n}$ = $\frac{1}{a^n}$

Expressions with Exponents:

Exponent terms along with numbers and other exponents are called as expressions with exponents.

Like $2x^3$, or  $x^5\ y^2$ etc.

To simplify expression with exponents we need to do the following rules

$a^m \times a^n$ = $a^{m + n}$

$\frac{a^m }{ a^n}$ = $a^{m-n}$

$(a^m)^n$ = $a^{m \times n}$

$(ab)^m$ = $a^m \times b^m$

 Related Calculators Calculating Exponents Fractional Exponents Calculator Adding Exponents Calculator Dividing Exponents Calculator

## Examples

Example 1:

Simplify $3a^3 \times 5a^7$

Solution:

$15 a^{3+7}$ = $15 a^{10}$
Example 2:

Simplify $(x^3 y^5 z^2)^2$ divided by $(xy - 1)^2$

Solution: Using the rule $(abc)^n$ = $a^n \times b^n \times c^n$ and $(a^m)^n$ = $a^{m \times n}$

$(x^3 y^5 z^2)^2$  = $(x^3)^2\ (y^5)^2\ (z^2)^2$ = $x^6y^{10}z^4$

$(xy^{-1})^2$ = $x^2y^{-2}$

Now we have to divide

$\frac{(x^6y^{10}z^4)}{( x^2y^{-2})}$

= $(x^{6-2})\ (y^{10-(-2)})\ (z^4)$

= $x^ 4y^{12} z^4\ \leftarrow\ Answer$
Example 3:

Write  $x \times x \times x \times x \times x$ as an exponential expression.

Solution:

$x \times x \times x \times x \times x$ = $x^5$
Example 4:

Simplify $\frac{(p^{-2}q^3)}{(p^{-5}q^9)}$

Solution:

Using $\frac{a^m }{a^n}$ = $a^{m – n}$

We get

$\frac{(p^{-2}q^3)}{(p^{-5}q^9)}$

= $p^{-2-(-5)}\ q^{3-9}$

= $p^{-2+5}\ q^{-6}$

= $p^3\ q^{-6}$

= $\frac{p^3}{q^6}$
Example 5:

Analyzing an exponents expression, state, the base and the exponent of the following exponent expression $125(1.05)^4$

Solution:

The base of the expression is $1.05$ and the exponent is $4$.

The above example is an application of exponent expression. These are used in problems of compound interest, growth and decay based problems. A problem could also be made “an investment of $\$125$grows at the rate of$5 \%$per year compounded; write an expression what the value of the investment after$\4 years.”  $125(1.05)^4$  would work as a solution

 More topics in  Exponents Properties of Exponents Exponent Rules Laws of Exponents Adding Exponents Subtracting Exponents Multiplying Exponents Dividing Exponents Fractional Exponents Negative Exponents Exponents with Variables Complex Exponents Integer Exponents Simplifying Exponents
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