Definition of x^{-n}
Apply the definition of negative exponents: x^(-n) equals 1/x^n, converting negative exponents to positive by moving the base across the fraction bar without changing its value.
Key Concepts
If n is any real number and x is any real number that is not zero, $$x^{ n} = \frac{1}{x^n}.$$ This definition says that when an exponential expression is written in reciprocal form, the sign of the exponent must be changed.
To simplify $5^{ 2}$, move the base to the denominator and make the exponent positive: $5^{ 2} = \frac{1}{5^2} = \frac{1}{25}$. The negative sign matters! $ 4^{ 2} = \frac{1}{16}$, but with parentheses, $( 4)^{ 2} = \frac{1}{( 4)^2} = \frac{1}{16}$. If the negative exponent is in the denominator, move it to the numerator: $\frac{1}{3^{ 2}} = 3^2 = 9$.
A negative exponent is a secret signal! It doesn't make the number negative. It just tells the base to move to the other side of the fraction bar, where its exponent then becomes positive. Flip it and switch the sign!
Common Questions
What does a negative exponent mean mathematically?
A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. By definition x^(-n) = 1/x^n for any nonzero x. For example 2^(-3) = 1/2^3 = 1/8, not -8.
How do you simplify an expression with negative exponents?
Move any factor with a negative exponent across the fraction bar, changing the sign of the exponent to positive. For (3x^(-2)y)/z^(-1), relocating the negative-exponent factors gives 3yz/x^2.
Why does a negative exponent not make the result negative?
Negative exponents represent division, not negation. The exponent sign controls placement in numerator or denominator. The sign of the base value determines whether the result is positive or negative. 2^(-3) = 1/8 is positive; (-2)^(-3) = -1/8 is negative.