The big idea: When the base stays the same, multiplication adds powers and division subtracts powers.
- same base
- powers to combine
Worked example
Simplify (a) x3 × x5 (b) y7 ÷ y2
Step by step
- Add powers when multiplying.
- Subtract powers when dividing.
Final answer
x8 and y5
Common mistake: Do not multiply the bases here. The base stays the same; only the powers change.
The big idea: A power raised to another power means multiply the powers. A bracket raised to a power applies to every factor inside.
- multiply the powers
Worked example
Simplify (a) (x4)3 (b) (2a)3
Step by step
- Multiply powers.
- Raise each factor inside the bracket.
Final answer
x12 and 8a3
Slow down on brackets: If a whole bracket is raised to a power, make sure the power applies to every factor inside the bracket.
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| Expression | Meaning |
|---|---|
| a0 | 1, provided a ≠ 0 |
| a-n | 1/an |
Worked example
Simplify (a) 50 (b) x-3
Step by step
- Any non-zero number to the power 0 equals 1.
- A negative power means reciprocal.
Final answer
1 and 1/x3
Why negative powers matter: A negative exponent does not make the answer negative. It tells you to write the term as a reciprocal.
Classic trap: x-2 is not -x2. It means 1/x2.
The big idea: In longer expressions, simplify step by step using the exponent laws in a sensible order instead of trying to do everything at once.
Worked example
Simplify \dfrac{(x3)2 \times x4}{x5}.
Step by step
- Power of a power first.
- Then multiply same base terms.
- Then divide.
Final answer
x5
Best habit: Write one clean line for each exponent law you use. This reduces sign mistakes and is easier to follow under exam pressure.