Exponent Laws: Powers, Negative and Fractional Exponents
Build from repeated multiplication to roots and reciprocals, one rule at a time.
What a power means
An exponent tells you how many copies of a base to multiply. In $3^4$, the base is 3 and the exponent is 4: $3^4=3[[times]]3[[times]]3[[times]]3=81$. This lesson starts with Year 7 skills and finishes with the negative and fractional powers often met by Years 9–10.
Start with powers
Work out the power before other multiplication: $2^3[[times]]5=8[[times]]5=40$. Brackets change the base: $(-2)^4=16$, but $-2^4=-16$ because the minus sign is outside the power.
Worked example: evaluate a power
Step 1
In $4^3$, multiply three copies of 4: $4[[times]]4[[times]]4$.
Step 2
$4^3=64$.
Combine powers
These rules use the same base. When you multiply, add exponents. When you divide, subtract exponents. Raising a power to another power multiplies the exponents.
$a^m[[times]]a^n=a^{m+n}$
$a^m[[div]]a^n=a^{m-n}$ for $a[[neq]]0$
$(a^m)^n=a^{mn}$
Worked example: multiply and divide
Step 1
$2^5[[times]]2^3=2^{5+3}=2^8$.
Step 2
$2^8[[div]]2^4=2^{8-4}=2^4=16$.
Step 3
For $(3^2)^3$, multiply the exponents: $3^{2[[times]]3}=3^6=729$.
A power distributes across multiplication and division: $(ab)^n=a^nb^n$ and $(a[[div]]b)^n=a^n[[div]]b^n$ when $b[[neq]]0$. For instance, $(2[[times]]3)^2=2^2[[times]]3^2=36$. Do not add the exponents of different bases: $2^3[[times]]3^2=8[[times]]9=72$.
Zero, negative and fractional powers
For any nonzero $a$, $a^0=1$. A negative exponent gives a reciprocal, not a negative answer: $a^{-n}=[[frac{1}{a^n}]]$. For example, $2^{-3}=[[frac{1}{2^3}]]=[[frac{1}{8}]]$.
Worked example: a negative power of a fraction
Step 1
Reciprocate the nonzero base: $([[frac{2}{3}]])^{-1}=[[frac{3}{2}]]$.
Step 2
Check: $[[frac{2}{3}]][[times]][[frac{3}{2}]]=1$.
In $a^{[[frac{m}{n}]]}$, the denominator $n$ tells you to take the $n$th root, then the numerator $m$ tells you which power to take. Use a nonnegative base for even roots in real-number questions.
$a^{[[frac{1}{n}]]}=[[sqrt[n]{a}]]$
$a^{[[frac{m}{n}]]}=([[sqrt[n]{a}]])^m$
Worked example: a fractional power
Step 1
$27^{[[frac{2}{3}]]}$ means take the cube root of 27: $[[sqrt[3]{27}]]=3$.
Step 2
Square the result: $3^2=9$.
Step 3
With a negative exponent, $27^{-[[frac{2}{3}]]}=[[frac{1}{9}]]$.
Check before you answer
- Keep the base unchanged when combining its powers.
- A zero power is 1 only when the base is nonzero; $0^0$ is undefined here.
- A negative exponent is a reciprocal, not a minus sign.
- For a fractional exponent, take the root named by the denominator.
Practice Problems
Try these in order. Each answer shows a method; the linked practice set generates new questions every time.
Practice 1
Work out $5^3$.
Show answer
$5^3=5[[times]]5[[times]]5=125$.
Practice 2
Work out $2^4[[times]]2^3$.
Show answer
$2^{4+3}=2^7=128$.
Practice 3
Work out $3^5[[div]]3^2$.
Show answer
$3^{5-2}=3^3=27$.
Practice 4
Work out $(2^3)^2$.
Show answer
$2^{3[[times]]2}=2^6=64$.
Practice 5
Work out $(2[[times]]5)^3$.
Show answer
$(10)^3=1000$; equivalently $2^3[[times]]5^3=1000$.
Practice 6
Work out $7^0$.
Show answer
$7^0=1$ because the base is nonzero.
Practice 7
Work out $4^{-2}$.
Show answer
$4^{-2}=[[frac{1}{4^2}]]=[[frac{1}{16}]]$.
Practice 8
Work out $([[frac{3}{5}]])^{-1}$.
Show answer
$([[frac{3}{5}]])^{-1}=[[frac{5}{3}]]$.
Practice 9
Work out $64^{[[frac{1}{2}]]}$.
Show answer
Take the square root: $[[sqrt{64}]]=8$.
Practice 10
Work out $8^{[[frac{2}{3}]]}$.
Show answer
Take the cube root first: $[[sqrt[3]{8}]]=2$; then $2^2=4$.
Practice 11
Work out $16^{-[[frac{3}{4}]]}$.
Show answer
Take the fourth root: $[[sqrt[4]{16}]]=2$. Cube it to get 8, then reciprocate: $[[frac{1}{8}]]$.
Interactive Quiz
Choose one answer for each question, then check your understanding.