To work with negative exponents, the key concept is that a negative exponent means taking the reciprocal of the base raised to the corresponding positive exponent. In other words:
- For any nonzero number aaa and positive integer nnn,
a−n=1ana^{-n}=\frac{1}{a^n}a−n=an1
This means instead of multiplying aaa by itself nnn times, you divide 1 by aaa multiplied by itself nnn times
How to Simplify Negative Exponents
- Rewrite the expression using the reciprocal:
Change the base with the negative exponent to its reciprocal and make the exponent positive.
Example:
2−4=124=1162^{-4}=\frac{1}{2^4}=\frac{1}{16}2−4=241=161
- If the negative exponent is in the denominator, move it to the numerator:
Example:
1a−n=an\frac{1}{a^{-n}}=a^na−n1=an
- For fractions with negative exponents, flip the fraction and change the exponent to positive:
Example:
(45)−3=(54)3\left(\frac{4}{5}\right)^{-3}=\left(\frac{5}{4}\right)^3(54)−3=(45)3
- Multiplying with negative exponents:
If bases are the same, add the exponents (including negatives).
Example:
a−m×a−n=a−(m+n)a^{-m}\times a^{-n}=a^{-(m+n)}a−m×a−n=a−(m+n)
- Dividing with negative exponents:
Subtract the exponents.
Example:
ama−n=am−(−n)=am+n\frac{a^m}{a^{-n}}=a^{m-(-n)}=a^{m+n}a−nam=am−(−n)=am+n
- With variables, apply the same rules:
Move the variable with a negative exponent to the opposite part of the fraction to make the exponent positive
Summary
- Negative exponent = reciprocal with positive exponent
- Move terms with negative exponents across the fraction bar to change the sign of the exponent
- Apply normal exponent rules (multiply, divide) after converting negative exponents to positive
This approach simplifies expressions involving negative exponents efficiently and consistently.
