Rational Exponents
Rational exponents (second instance) is a Grade 7 algebra skill from Yoshiwara Intermediate Algebra where x^(m/n) = (nth root of x)^m unifies radical and exponential expressions. Students practice converting, simplifying, and applying all standard exponent laws to fractional exponents.
Key Concepts
Property For a positive base $a$ and a rational exponent $\frac{m}{n}$ where $n \neq 0$: $$a^{m/n} = (a^{1/n})^m = (a^m)^{1/n}$$ To compute $a^{m/n}$, you can either take the $n$th root of $a$ first and then raise it to the $m$th power, or raise $a$ to the $m$th power and then take the $n$th root. The denominator of the exponent is the root, and the numerator is the power.
Examples To evaluate $64^{2/3}$, we can take the cube root of 64 first, which is 4, and then square it: $(64^{1/3})^2 = 4^2 = 16$.
For $ 8^{4/3}$, the exponent applies only to the 8. We find $(8^{1/3})^4 = 2^4 = 16$, so the expression equals $ 16$.
Common Questions
What does x^(m/n) mean?
x^(m/n) = (nth root of x)^m. The denominator n is the root index, and the numerator m is the power applied after taking the root.
How do you simplify 27^(2/3)?
27^(2/3) = (cube root of 27)^2 = 3^2 = 9.
Can you apply the power rule to rational exponents?
Yes. (x^(m/n))^p = x^(mp/n). All standard exponent rules extend to rational exponents.
What is x^(-1/2)?
x^(-1/2) = 1/x^(1/2) = 1/sqrt(x). The negative sign means reciprocal.