4.01 Review: 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
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 $20m^6\div5m^{13}\times9m^2$20m6÷5m13×9m2, expressing your answer in positive exponential form.
Think: Let's express this as a fraction so the powers are on the numerator and the denominator for easy comparison.
Do:
| $\frac{20m^6}{5m^{13}}\times9m^2$20m65m13×9m2 | $=$= | $\frac{4}{m^7}\times9m^2$4m7×9m2 | Simplify the fraction using the quotient property |
| $=$= | $\frac{36m^2}{m^7}$36m2m7 | Simplify the multiplication | |
| $=$= | $36m^{-5}$36m−5 | Use the quotient property - this step is sometimes omitted | |
| $=$= | $\frac{36}{m^5}$36m5 | Write as a positive exponent |
Practice questions
Question 5
Simplify $\frac{\left(x^2\right)^6}{\left(x^2\right)^2}$(x2)6(x2)2
Question 6
Simplify $\left(u^9\cdot u^5\div u^{19}\right)^2$(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.