Rational exponents

Higher Level

The curve is 16^p. The whole-number powers are marked. Slide p and watch where the fractions land.

p = 0.5

416^p

√16in words

Halfway in the exponent is not halfway in the value.

The rule

x^a/b = ^b√(x^a) = (^b√x)^a, the bottom is the root, the top is the power, and either order gives the same answer. Taking the root first keeps the numbers small, which is almost always the easier route by hand.

16^1/2= √16 = 4 32^3/5= (⁵√32)³ = 2³ = 8 8^2/3= (³√8)² = 2² = 4 16^−1/2= 1√16 = 0.25 5^1/2 × 5^1/3= 5^5/6 add the fractions

A negative power never makes a negative answer. It makes a reciprocal. 16^−1/2 is a quarter, not −4, and the curve above never dips below the axis however far left you slide.

Why the laws still work

Nothing new is needed. The same three rules from 1.5 apply, with fractions in place of integers.

multiply→ add the exponents: ½ + ⅓ = ⅝ divide→ subtract: 64^1/2 ÷ 64^1/3 = 64^1/6 = 2 power of a power→ multiply: (x^2/3)^3/4 = x^1/2

Worked example

The surface area of a sphere is 4πr² and its volume is ⁴⁄₃πr³. A ball has volume 2304π cm³.

Find its surface area.

⁴⁄₃πr³ = 2304πThe pi cancels, which is why the question was written with pi in it.
r³ = 2304 × ³⁄₄ = 1728Multiply by the reciprocal. 1728 is a cube, which is a hint the numbers were chosen.
r = 1728^1/3 = 12 cmThe fractional power is a root. Not 1728 divided by 3.
A = 4π × 12²Now substitute into the area formula.
= 576π cm², about 1810 cm²Leave it as 576π unless asked for a decimal: that is the exact answer, and rounding early would lose accuracy for no reason.

Your turn

1. Find 16^1/2.

2. Find 8^2/3.

3. Find 16^−1/2.

Where the marks go

Reading the fraction correctly: bottom is the root, top is the power.

Negative powers as reciprocals, keeping the answer positive.

Leaving an exact answer exact when the question asks for one. 2√3 is a finished answer; 3.46 is a rounded one.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

log₁₀ 50 lies between which two whole numbers?

Logs, estimated not computed. 50 is between 10 and 100, so the log is between 1 and 2. It is 1.699, and the estimate catches a mistyped entry.
Which method?

A sequence is 10, 6, 3.6, 2.16. Find the sum to infinity.

Geometric with r = 0.6. 10/0.4 = 25. Check |r| < 1 before using the formula.
Which method?

Simplify 64^(1/2) ÷ 64^(1/3).

Subtract the exponents: 64^(1/6) = 2. Fractions subtract exactly as integers do.

The same idea elsewhere

5.3, differentiating powers: The power rule needs rational indices, which is why this comes first.
1.1, standard form: Both are about what a power actually counts.

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