A reliable method
Rewrite expressions using one base where possible. Add exponents in a product, subtract them in a quotient with a nonzero base, and multiply them for a power of a power.
The common trap
An even root denotes a nonnegative principal value. The expression √(x²) equals |x| for real x, rather than always equaling x.
Check and transfer
Keep restrictions visible when simplifying. Cancelling powers of x in a denominator assumes x is not zero; algebraic appearance alone does not remove that original condition.
Worked comparison: identify the structure
For nonzero x, x⁷/x³=x⁴, while (x⁷)³=x²¹. Subtract exponents for a quotient, multiply them for nested powers. Testing x=2 gives 128/8=16 in the first case, which agrees with 2⁴ and refutes the added-exponent answer 2¹⁰.
Transfer and explain
Evaluate 4⁻²=1/16, not −16. Then √((−4)²)=4: a negative exponent changes a reciprocal relationship, whereas a principal even root chooses a nonnegative value.
For x ≠ 0, x^9/x^5 = x^k. What is k?
- The bases are equal, so subtract exponents in the quotient.
- The nonzero restriction belongs to the original denominator.
- The requested value is 4. Substitute or reverse the operation to check this result.
Compare the other answer choices
Cancel the entire denominator without changing the numerator exponent.
Multiply the exponents as if the expression were a power of a power.
Divide the exponents rather than subtracting them.
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.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference