3.01 Laws of exponents
Let's review the laws of exponents. It's important to remember the order of operations when we're simplifying these expressions.
- The product of powers property: $a^m\times a^n=a^{m+n}$am×an=am+n
- The quotient of powers property: $a^m\div a^n=a^{m-n}$am÷an=am−n
- The zero exponent property: $a^0=1$a0=1
- The power of a power property: $\left(a^m\right)^n=a^{mn}$(am)n=amn
- The negative exponent definition: $a^{-m}=\frac{1}{a^m}$a−m=1am
- Fractional exponents: $a^{\frac{m}{n}}=\sqrt[n]{a^m}$amn=n√am
A question may have any combination of laws of exponents. We just need to simplify it step by step, making sure we follow the order of operations.
Worked examples
Question 1
Simplify: $p^7\div p^3\times p^5$p7÷p3×p5
Think: We need to apply the exponent division and exponent multiplication laws.
Do:
| $p^7\div p^3\times p^5$p7÷p3×p5 | $=$= | $p^{7-3+5}$p7−3+5 |
| $=$= | $p^9$p9 |
Reflect: We can choose to do this in more steps by first doing $p^{7-3}\times p^5=p^4\times p^5$p7−3×p5=p4×p5 and then getting our final answer of $p^9$p9.
Question 2
Simplify: $\frac{\left(u^{x+3}\right)^3}{u^{x+1}}$(ux+3)3ux+1
Think: We need to simplify the numerator using the power of a power property, then apply the quotient property.
Do:
| $\frac{\left(u^{x+3}\right)^3}{u^{x+1}}$(ux+3)3ux+1 | $=$= | $\frac{u^{3\left(x+3\right)}}{u^{x+1}}$u3(x+3)ux+1 |
Simplify the numerator using the power of a power property |
| $=$= | $\frac{u^{3x+9}}{u^{x+1}}$u3x+9ux+1 |
Apply the distributive property |
|
| $=$= | $u^{3x+9-\left(x+1\right)}$u3x+9−(x+1) |
Use the quotient property and subtract the powers |
|
| $=$= | $u^{3x+9-x-1}$u3x+9−x−1 |
Simplify by collecting the like terms |
|
| $=$= | $u^{2x+8}$u2x+8 |
Question 3
Express $\left(4^p\right)^4$(4p)4 with a prime number base in exponential form.
Think: We could express $4$4 as $2^2$22 which has a prime number base.
Do:
| $\left(4^p\right)^4$(4p)4 | $=$= | $4^{4p}$44p |
Use the power of a power property |
| $=$= | $\left(2^2\right)^{4p}$(22)4p |
Use the fact that $4=2^2$4=22 |
|
| $=$= | $2^{8p}$28p |
Use the power of a power property |
Reflect: This skill will become increasingly important as we look at simplifying expressions with related bases such as $2^{3p}\times\left(4^p\right)^4$23p×(4p)4.
Question 4
Simplify $\sqrt{20m^6}\div\sqrt[3]{8m^9}$√20m6÷3√8m9, assume $m>0$m>0.
Think: Let's express this as a fraction so the powers are on the numerator and the denominator for easy comparison. Also, we will write the radicals as fractional exponents to allow for further simplification.
Do:
| $\sqrt{20m^6}\div\sqrt[3]{8m^9}$√20m6÷3√8m9 | $=$= | $\frac{\left(20m^6\right)^{\frac{1}{2}}}{\left(8m^9\right)^{\frac{1}{3}}}$(20m6)12(8m9)13 |
Rewrite the radicals as fractional exponents |
| $=$= | $\frac{20^{\frac{1}{2}}\left(m^6\right)^{\frac{1}{2}}}{8^{\frac{1}{3}}\left(m^9\right)^{\frac{1}{3}}}$2012(m6)12813(m9)13 |
Use the power of a product property |
|
| $=$= | $\frac{20^{\frac{1}{2}}m^3}{8^{\frac{1}{3}}m^3}$2012m3813m3 |
Use the power of a power |
|
| $=$= | $\frac{20^{\frac{1}{2}}}{8^{\frac{1}{3}}}$2012813 |
Use the quotient of powers property |
|
| $=$= | $\frac{\left(2^2\right)^{\frac{1}{2}}\times5^{\frac{1}{2}}}{\left(2^3\right)^{\frac{1}{3}}}$(22)12×512(23)13 |
Rewrite bases to take fractional powers |
|
| $=$= | $\frac{2\sqrt{5}}{2}$2√52 |
Simplify fractional powers |
|
| $=$= | $\sqrt{5}$√5 |
Simplify |
Practice questions
Question 5
Simplify $\frac{\left(x^2\right)^6}{\left(x^2\right)^2}$(x2)6(x2)2
Question 6
Simplify $\left(u^9\times u^5\div u^{19}\right)^2$(u9×u5÷u19)2, expressing your answer in positive exponential form.
Question 7
Express $\left(5y^3\right)^{-3}$(5y3)−3 with a positive exponent.
Question 8
Evaluate $4^{\frac{3}{2}}$432.