An expression in the form $a^n$an is called an exponential expression. The letter $a$a is called the base and $n$n is called the exponent or index. Both the base and index can be a positive or negative integer or fraction.
The index laws
The product rule
The product rule applies when we have the product of multiple exponential expressions with the same base and states:
| $a^m\times a^n$am×an | $=$= | $a^{m+n}$am+n |
The quotient rule
The quotient rule applies when we have multiple exponential expressions with the same base divided by each other and states:
| $a^m\div a^n$am÷an | $=$= | $a^{m-n}$am−n |
The zero index rule
The zero index rule applies when any number or expression is raised to the power of $0$0 and states:
| $a^0$a0 | $=$= | $1$1 |
The power of a power rule
The quotient rule applies when we have multiple exponential expressions with the same base divided by each other and states:
| $\left(a^m\right)^n$(am)n | $=$= | $a^{mn}$amn |
The negative index rule
The negative index rule applies when we have an exponential expressions with a negative index and states:
| $a^{-m}$a−m | $=$= | $\frac{1}{a^m}$1am |
The fractional index rule
The fractional index rule applies when we have an exponential expression with with a fractional index and states:
| $a^{\frac{m}{n}}$amn | $=$= | $\sqrt[n]{a^m}$n√am |
The fractional base rule
The fractional base rule applies when we have a fractional base number raised to a power and states:
| $\left(\frac{a}{b}\right)^m$(ab)m | $=$= | $\frac{a^m}{b^m}$ambm |
Be careful when using index rules with numerical bases. Index laws still apply and the bases have to be the same! For example, if we have an expression such as $3^x\times9^x$3x×9x we can not apply any of these rules as $3\ne9$3≠9. However, if we can convert $9$9 into the same base of $3$3 we can proceed as follows:
| $3^x\times9^x$3x×9x | $=$= | $3^x\times\left(3^2\right)^x$3x×(32)x |
| $=$= | $3^x\times3^{2x}$3x×32x | |
| $=$= | $3^{x+2x}$3x+2x | |
| $=$= | $3^{3x}$33x |
Let's have a look at how to simplify (to simplest index form) some more complicated examples.
Worked example
Example 1
Simplify the expression $\frac{6^{p-4q}\times36^{2q+p}}{6^{5p-3q}}$6p−4q×362q+p65p−3q
Think: First make sure all terms have the same base and be careful with the order of operations when subtraction of groups of terms is concerned.
Do:
| $\frac{6^{p-4q}\times36^{2q+p}}{6^{5p-3q}}$6p−4q×362q+p65p−3q | $=$= | $\frac{6^{p-4q}\times\left(6^2\right)^{2q+p}}{6^{5p-3q}}$6p−4q×(62)2q+p65p−3q |
| $=$= | $\frac{6^{p-4q}\times6^{2\left(2q+p\right)}}{6^{5p-3q}}$6p−4q×62(2q+p)65p−3q | |
| $=$= | $\frac{6^{p-4q}\times6^{4q+2p}}{6^{5p-3q}}$6p−4q×64q+2p65p−3q | |
| $=$= | $\frac{6^{p-4q+4q+2p}}{6^{5p-3q}}$6p−4q+4q+2p65p−3q | |
| $=$= | $\frac{6^{3p}}{6^{5p-3q}}$63p65p−3q | |
| $=$= | $6^{3p-\left(5p-3q\right)}$63p−(5p−3q) | |
| $=$= | $6^{3p-5p+3q}$63p−5p+3q | |
| $=$= | $6^{3q-2p}$63q−2p |
Practice questions
question 1
Simplify the following, giving your answer with a positive index: $m^9\div m^5\times m^4$m9÷m5×m4
question 2
Simplify the following, giving your answer with a positive index: $y^{\frac{6}{5}}\times y^2$y65×y2
question 3
Simplify $\frac{b^3\div b^{-7}}{\left(b^{-4}\right)^{-4}}$b3÷b−7(b−4)−4, giving your answer without negative indices.
Question 4
Fill in the blanks to simplify the given expression.
$\sqrt{m^8}$√m8 $=$= $\left(m^8\right)^{\editable{}}$(m8) $=$= $m^{\editable{}\times\frac{1}{2}}$m×12 $=$= $m^{\editable{}}$m
Question 5
The expression $\sqrt[3]{a^2}$3√a2 can also be expressed in index form as $a^{\frac{x}{y}}$axy. What are the values of $x$x and $y$y?
$x$x $=$= $\editable{}$
$y$y $=$= $\editable{}$
Question 6
Simplify the following, writing without negative indices.
$7p^4q^{-8}\times4p^{-4}q^{-5}$7p4q−8×4p−4q−5
Question 7
Simplify the following, giving your answer with a single index:
$\frac{3^{-20b-4}\times9^{3b+2}}{81^{-3b-3}}$3−20b−4×93b+281−3b−3