topic badge
CanadaON
Grade 12

Review the properties of exponents

Lesson

Hey, remember exponents (or powers)? Those little numbers that float above other numbers or variables? Of course you do, so let's revise the various laws we've learnt about and practice using them!

Rule Recap
  • The product rule: $a^m\times a^n=a^{m+n}$am×an=am+n
  • The quotient rule: $a^m\div a^n=a^{m-n}$am÷​an=amn
  • The zero exponent rule:$a^0=1$a0=1
  • The power of a power rule: $\left(a^m\right)^n=a^{mn}$(am)n=amn
  • The negative exponent rule: $a^{-m}=\frac{1}{a^m}$am=1am
  • The fractional exponent rule: $a^{\frac{m}{n}}=\sqrt[n]{a^m}$amn=nam
  • The fractional base rule: $\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}$(ab)m=ambm

Now let's put them to the test in the following problems! Remember to be mindful of your order of operations when doing these problems.

 

Examples

QUESTION 1

Express in the simplest exponential form: $\left(\frac{2^0}{4^2}\right)^2$(2042)2

Think: We have too many powers so let's first see if you can express it without the brackets. Also, exponential form means we need to leave the powers rather than evaluate them.

Do:

$\left(\frac{2^0}{4^2}\right)^2$(2042)2 $=$= $\frac{2^{0\times2}}{4^{2\times2}}$20×242×2
  $=$= $\frac{2^0}{4^4}$2044
  $=$= $\frac{1}{4^4}$144

 

Question 2

Simplify: $k^{\frac{1}{2}}\times\left(-2k^3\right)^2$k12×(2k3)2

Think: When a negative number/variable is squared, it becomes positive again

Do:

$k^{\frac{1}{2}}\times\left(-2k^3\right)^2$k12×(2k3)2 $=$= $k^{\frac{1}{2}}\times\left(-2\times k^3\right)^2$k12×(2×k3)2
  $=$= $k^{\frac{1}{2}}\times\left(-2\right)^2\times k^{3\times2}$k12×(2)2×k3×2
  $=$= $k^{\frac{1}{2}}\times4\times k^6$k12×4×k6
  $=$= $4k^{\frac{1}{2}+6}$4k12+6
  $=$= $4k^{\frac{13}{2}}$4k132

 

question 3

Express in positive exponential form: $\frac{s^{-4}}{s^2\times s^{-5}}$s4s2×s5

Think: Simplify all powers before trying to convert negative exponents into their positive counterparts

Do:

$\frac{s^{-4}}{s^2\times s^{-5}}$s4s2×s5 $=$= $\frac{s^{-4}}{s^{2+\left(-5\right)}}$s4s2+(5)
  $=$= $\frac{s^{-4}}{s^{-3}}$s4s3
  $=$= $s^{-4-\left(-3\right)}$s4(3)
  $=$= $s^{-1}$s1
  $=$= $\frac{1}{s}$1s

 

More Worked Examples

Question 4

Simplify the following, giving your answer with a positive exponent: $2^2\times2^2$22×22

Question 5

Simplify $p^7\div p^3\times p^5$p7÷​p3×p5.

Question 6

Simplify the following, writing without negative exponents.

$7p^4q^{-8}\times4p^{-4}q^{-5}$7p4q8×4p4q5

What is Mathspace

About Mathspace