SJMATHTUBE • LAWS OF EXPONENTS (INDICES)
Laws of Exponents
These rules help you simplify expressions with powers. Most rules need the same base.
Important: Assume bases are non-zero where division is involved (so we do not divide by 0). For roots, assume the expression is defined.
1
Product of powers (same base)
When multiplying powers with the same base, add the exponents.
\( a^{m}\cdot a^{n}=a^{m+n} \)
Example: \(2^{3}\cdot 2^{4}=2^{7}\)
2
Quotient of powers (same base)
When dividing powers with the same base, subtract the exponents.
\( \frac{a^{m}}{a^{n}}=a^{m-n}\;\;\text{(}a\neq 0\text{)} \)
Example: \( \frac{5^{6}}{5^{2}}=5^{4} \)
3
Power of a power
When a power is raised to another power, multiply the exponents.
\( (a^{m})^{n}=a^{mn} \)
Example: \( (3^{2})^{4}=3^{8} \)
4
Power of a product
A power outside brackets applies to every factor inside.
\( (ab)^{n}=a^{n}b^{n} \)
Example: \( (2x)^{3}=2^{3}x^{3} \)
5
Power of a quotient
A power outside a fraction applies to numerator and denominator.
\( \left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}\;\;\text{(}b\neq 0\text{)} \)
Example: \( \frac{3^{2}}{5^{2}}=\frac{9}{25} \)
6
Zero exponent
Any non-zero base raised to 0 equals 1.
\( a^{0}=1\;\;\text{(}a\neq 0\text{)} \)
Example: \( 7^{0}=1 \)
7
Negative exponent
A negative exponent means reciprocal.
\( a^{-n}=\frac{1}{a^{n}}\;\;\text{(}a\neq 0\text{)} \)
Example: \( 2^{-3}=\frac{1}{2^{3}} \)
8
Fractional exponents (roots)
Fractional powers represent roots.
\( a^{1/n}=\sqrt[n]{a} \)
\( a^{m/n}=\sqrt[n]{a^{m}} \)
Example: \( 9^{1/2}=\sqrt{9}=3 \)
9
Power of 1
Exponent 1 leaves the base unchanged.
\( a^{1}=a \)
Example: \( x^{1}=x \)
10
Special base facts
These are common results worth remembering.
\( 1^{n}=1 \;\;\;\; 0^{n}=0 \;\; \text{(}n>0\text{)} \)
\( 0^{0} \text{ is undefined} \)
11
Same exponent, different bases
You can combine only when the exponent is the same.
\( a^{n}b^{n}=(ab)^{n} \)
Note: Addition is different: \( a^{m}+a^{n} \) cannot be combined.
12
The “same base” rule
Most exponent laws need the same base. If you see a plus sign, do not merge exponents.
\( \textbf{Works: } 2^{3}\cdot 2^{4}=2^{7} \)
\( \textbf{Does not work: } 2^{3}+2^{4}\neq 2^{7} \)
Common mistake: Exponent rules do not work for addition or subtraction.
For example, \(a^{m}+a^{n}\) does not become \(a^{m+n}\).