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 $a^{-n}$ = $\frac{1}{a^n}$

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 |

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

Solution:

$15 a^{3+7}$ = $15 a^{10}$

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

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

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

Now we have to divide

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

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

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

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

$x \times x \times x \times x \times x$ = $x^5$

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

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

We get

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

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

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

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

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

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 βan investment of $\$125$ grows at the rate of $5 \%$ per year compounded; write an expression what the value of the investment after $\$$4 years.β $125(1.05)^4$ would work as a solution