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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.

Laws of exponents
  • 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=amn
  • 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: $a^{-m}=\frac{1}{a^m}$am=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.


$p^7\div p^3\times p^5$p7÷​p3×p5 $=$= $p^{7-3+5}$p73+5
  $=$= $p^9$p9

Reflect: We can choose to do this in more steps by first doing $p^{7-3}\times p^5=p^4\times p^5$p73×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.


$\frac{\left(u^{x+3}\right)^3}{u^{x+1}}$(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+9x1 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.


$\left(4^p\right)^4$(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.


$\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}$36m5 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 $\left(u^9\times u^5\div u^{19}\right)^2$(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.

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