Choose a representation
For a positive base, interpret fractional powers through roots. Add exponents for multiplication and subtract for division. To add rational terms, use a shared denominator.
Worked example
For positive x=16, what is x raised to the power 3/2? The denominator 2 asks for the positive square root. Raise that root to the third power. Substitute the given numbers: 4^3 = 64. The requested value is 64.
Contrast a wrong operation
In that example, 4 follows this error: Stop after taking the square root. The correct requested value is 64. Explain which quantity the mistaken method is actually calculating.
Connect a second form
When (2 x+4)(x²+6 x+1) is expanded, what is the coefficient of x²? Identify all products that produce exactly x². Add 2 x times 6 x and 4 times x²; other products have different degrees. Substitute the given numbers: 2 × 6+4 = 16. The requested value is 16.
Transfer and check
For positive x=25, what is x raised to the power 3/2? Pause before checking. Answer: 125. The denominator 2 asks for the positive square root. Raise that root to the third power. Substitute the given numbers: 5^3 = 125.
Boundaries of the method
A negative exponent takes a reciprocal, not a negative value. Multiplication across brackets distributes to all terms. A cancelled factor does not make a previously forbidden input valid. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
For positive x=16, what is x raised to the power 3/2?
- The denominator 2 asks for the positive square root.
- Raise that root to the third power.
- Substitute the given numbers: 4^3 = 64.
- The requested value is 64.
Compare the other answer choices
Stop after taking the square root.
Cube the original x without taking its square root.
Take a reciprocal even though the exponent is positive.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications