Negative Exponents on Fractions
Grade 8 math lesson on negative exponents applied to fractions, using the rule that a negative exponent means taking the reciprocal and raising it to the positive power. Students learn to simplify expressions like (a/b)^(-n) = (b/a)^n.
Key Concepts
Property To handle a negative exponent on a fraction, flip the fraction to its reciprocal and make the exponent positive. $$ \left(\frac{x}{y}\right)^{ n} = \left(\frac{y}{x}\right)^n $$.
Examples $(\frac{1}{3})^{ 2}$ becomes $3^2$, which equals $9$. $(\frac{1}{2})^{ 3}$ becomes $2^3$, which equals $8$. $(\frac{3}{2})^{ 1}$ becomes $(\frac{2}{3})^1$, which equals $\frac{2}{3}$.
Explanation Think of a negative exponent as a secret 'flip' command! When a fraction gets this command, it does a somersault—the bottom number goes to the top and the top goes to the bottom. The exponent then becomes positive and happy.
Common Questions
What does a negative exponent on a fraction mean?
A negative exponent on a fraction means taking the reciprocal and using a positive exponent. So (a/b)^(-n) = (b/a)^n. For example, (2/3)^(-2) = (3/2)^2 = 9/4.
How do you simplify a fraction with a negative exponent?
Flip the fraction (take the reciprocal) and change the exponent to positive. Then evaluate the positive power normally. For (3/4)^(-3): flip to (4/3)^3 = 64/27.
Why does a negative exponent mean taking the reciprocal?
Negative exponents are defined so that the exponent rules remain consistent. x^(-1) = 1/x because x^1 x x^(-1) must equal x^0 = 1, requiring x^(-1) = 1/x. This extends to all negative exponents.
What is the general rule for negative exponents?
The general rule is x^(-n) = 1/x^n for any non-zero base x. For a fraction (a/b)^(-n) = (b/a)^n. This rule allows you to rewrite any negative exponent expression with a positive exponent.