Solving Equations with Rational Exponents
Solving equations with rational exponents is a Grade 7 math skill from Yoshiwara Intermediate Algebra where students isolate a variable raised to a fractional exponent by raising both sides to the reciprocal power. Care is taken with even and odd roots to handle both positive and negative solutions.
Key Concepts
Property To solve an equation involving a variable raised to a rational exponent, first isolate the power. Then, raise both sides of the equation to the reciprocal of that exponent.
If $X^{m/n} = C$, then raise both sides to the power of $\frac{n}{m}$: $$(X^{m/n})^{n/m} = C^{n/m}$$ $$X = C^{n/m}$$ This process works because $(X^{m/n})^{n/m} = X^{(m/n) \cdot (n/m)} = X^1 = X$.
Examples To solve $x^{3/2} = 64$, raise both sides to the reciprocal power of $\frac{2}{3}$: $(x^{3/2})^{2/3} = 64^{2/3}$, which simplifies to $x = (\sqrt[3]{64})^2 = 4^2 = 16$.
Common Questions
How do you solve an equation with a rational exponent?
Isolate the term with the fractional exponent, then raise both sides to the reciprocal power. For example, x^(2/3) = 4 → x = 4^(3/2) = 8.
Why must you be careful with even-index roots?
When raising to a power with an even denominator, the result could be positive or negative. Always consider ± when the index is even.
How do you solve x^(3/2) = 8?
Raise both sides to the 2/3 power: x = 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.
What is the relationship between rational exponents and radical equations?
An equation with a rational exponent x^(m/n) = c is equivalent to the radical equation (nth root of x)^m = c.