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!
- 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=am−n
- 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: $a^{-m}=\frac{1}{a^m}$a−m=1am
- The fractional exponent rule: $a^{\frac{m}{n}}=\sqrt[n]{a^m}$amn=n√am
- 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: $\frac{s^{-4}}{s^2\times s^{-5}}$s−4s2×s−5
Think: Simplify all powers before trying to convert negative exponents into their positive counterparts
Do:
| $\frac{s^{-4}}{s^2\times s^{-5}}$s−4s2×s−5 | $=$= | $\frac{s^{-4}}{s^{2+\left(-5\right)}}$s−4s2+(−5) |
| $=$= | $\frac{s^{-4}}{s^{-3}}$s−4s−3 | |
| $=$= | $s^{-4-\left(-3\right)}$s−4−(−3) | |
| $=$= | $s^{-1}$s−1 | |
| $=$= | $\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}$7p4q−8×4p−4q−5