A reliable method
Rewrite both sides with a common positive base when possible. List the repeating last-digit cycle and divide the exponent by its length. Keep odd and even exponents separate for negative bases.
The common trap
The principal square root is nonnegative. The equation x² = a² has two roots when a is nonzero; √(a²) equals |a|, not always a.
Make the connection
Test small exponents before generalizing a cycle. Track whether the expression contains a negative base in parentheses or a minus sign outside the power.
Contrast the cases
The symbol √25 means 5, while the equation x²=25 allows x=5 and x=−5. These are different requests. Powers of 3 have last digits 3,9,7,1, repeating every four: exponent 6 uses position 2 and ends in 9. A zero remainder selects the fourth position, not a digit of zero. For 4ˣ=2⁸, rewriting 4 as 2² gives 2x=8.
Transfer and explain
Transfer: find √((-7)²), solve x²=49, and find the last digit of 3⁸. Results are 7; ±7; and 1. Name the rule that distinguishes each task.
If 4ˣ = 2^10, what is x?
- Rewrite 4ˣ as 2^(2x).
- The common positive base implies 2x = 10.
- x = 5.
Compare the other answer choices
Subtracts exponents instead of solving the linear exponent equation.
Equates exponents before converting both bases.
Returns the exponential value rather than its exponent.
Divides by the base rather than its exponent.
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.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference