The big idea: When you multiply, divide or raise powers of the same base, you work with the exponents, not the big numbers.
For example, 2³ × 2⁴ = 2⁷ — just add the exponents.
See it on numbers
Simplify (x⁵ × x³) ÷ x².
Step by step
- Multiply — add the exponents.
- Divide — subtract the exponent.
Final answer
x⁶.
Same base only: These laws only work when the base is the same — you cannot combine 2³ × 3² this way.
And anything to the power 0 is 1: a⁰ = 1.
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Roots and reciprocals are powers too: A negative exponent means "one over"; a fractional exponent means a root — so you can turn roots and reciprocals into clean powers, and back.
See it — evaluate a fractional power
Evaluate 272/3 without a calculator.
Step by step
- The denominator is the root: take the cube root first.
- The numerator is the power: square it.
Final answer
272/3 = 9.
IB-style question — write as a single power
Write each as a power of x: u = 1/x³ v = ∛x w = x²√x
Step by step
- Bottom of a fraction → negative exponent.
- Cube root → exponent 1/3.
- √x = x1/2; multiplying means adding exponents.
Final answer
x⁻³, x1/3, x5/2.
IB-style question — solve for the base
Given that a2/3 = 4, find a.
Step by step
- To undo the power 2/3, raise both sides to the reciprocal 3/2.
- Square root first, then cube.
Final answer
a = 8.
Practice with real exam questions
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A hidden quadratic: A neat exam twist: an equation with aˣ and a²ˣ is secretly a quadratic, because a²ˣ = (aˣ)².
Substitute y = aˣ, solve, then solve back for x.
IB-style question — substitute and solve
Solve 4ˣ − 6(2ˣ) + 8 = 0.
Step by step
- Spot the hidden square: 4ˣ = (2²)ˣ = (2ˣ)². Let y = 2ˣ.
- Factorise the quadratic.
- Solve back for x (recall y = 2ˣ).
Final answer
x = 1 or x = 2.
Reject impossible values: A power like 2ˣ is always positive.
If the quadratic gives a negative or zero value of y, reject it — keep only y > 0 before solving back for x.