Choose a representation
For a positive base, interpret a fractional exponent as a root followed by a power. Convert a logarithmic equation to an exponential equation before isolating the unknown.
Worked example: follow the quantity
What is the positive real value of 27 raised to the power 2/3? Take the cube root because the exponent denominator is 3. Square that root because the numerator is 2. Substitute the given numbers: 3^2 = 9. Therefore the requested value is 9.
Contrast a plausible mistake
For that example, the tempting answer 1/9 follows this error: Treat the positive exponent as negative. The requested quantity is 9. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
If log base 3 of (x+2) equals 3, what is x? Rewrite as x+2 equal to the base cubed. Subtract 2 and check that x+2 is positive. Substitute the given numbers: 3^3-2 = 25. Therefore the requested value is 25.
Transfer before checking
What is the positive real value of 64 raised to the power 2/3? Try without viewing the answer. Check: 16. Take the cube root because the exponent denominator is 3. Square that root because the numerator is 2. Substitute the given numbers: 4^2 = 16.
Scope and conditions
In real-number work a logarithm needs positive argument, positive base other than one. Taking the negative of a power is different from taking its reciprocal. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
What is the positive real value of 27 raised to the power 2/3?
- Take the cube root because the exponent denominator is 3.
- Square that root because the numerator is 2.
- Substitute the given numbers: 3^2 = 9.
- Therefore the requested value is 9.
Compare the other answer choices
Stop after taking the cube root.
Multiply the cube root by 2 instead of squaring.
Treat the positive exponent as negative.
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.
ACT Mathematics content