A reliable method
Identify whether the operation is multiplication, division, or a power of a power before choosing a rule.
The common trap
The sum x² + x³ is not x⁵. The product x² · x³ is x⁵. An exponent rule does not combine a sum.
Make the connection
Rewrite radicals as fractional powers when the real-number domain permits it. Track restrictions such as a nonzero denominator.
Choose the rule from the operation
For x>0, x³·x⁴=x⁷ because there are seven factors of x. By contrast, (x³)⁴=x¹² because four groups each contain three factors. The expression x³+x⁴ cannot be combined using either rule; factoring gives x³(1+x). Testing x=2 distinguishes the sum 24 from 2⁷=128.
Check domain and cancellation
The quotient x⁷/x⁴=x³ requires x≠0 before cancellation. A simplified expression may appear defined at an input that the original excludes. Keep that restriction even after the denominator disappears. A negative exponent similarly encodes a reciprocal, rather than a negative value: 2⁻³=1/8.
Try, then check
If 3^t=7, find 3^(t+2) without finding t. Split the exponent to obtain 3^t·3²=7·9=63. Adding 2 to 7 confuses an exponent change with an output change. Now compare 3^(2t): it equals (3^t)²=49, so the two requests require different operations.
For x > 0, x^3 · x^4 = x^n. What is n?
- The bases are the same and the powers are multiplied.
- Add the exponents rather than multiplying them.
- n = 3 + 4 = 7.
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.
Official mathematics skillsSiyavula Grade 10 mathematics