Rational exponents in exponential functions

Three quick questions from earlier lessons.

A bacteria population doubles every day. By what factor does it grow in half a day?

Hint: whatever the factor $f$ is, growing for two half-days must double the population, so $f \cdot f = 2$ .

Exponents that are fractions are roots.

b^{\frac{1}{n}} = \sqrt[n]{b}

and

b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m

So the half-day factor is $2^{\frac12} = \sqrt{2} \approx 1.41$ . A negative exponent means a reciprocal: $b^{-r} = \dfrac{1}{b^{r}}$ .

Evaluate $8^{\frac{2}{3}}$ .

8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2

The denominator 3 means a cube root; the numerator 2 means a square.

= 2^2

The cube root of 8 is 2.

= 4

Square it.

Your turn: evaluate $27^{\frac{2}{3}}$ . Fill in the missing line.

27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2

Cube root, then square.

= 9

3² = 9.

Three questions to check you have got it.

Rational exponents

$b^{\frac{1}{n}} = \sqrt[n]{b}$ ; $b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m$ .
$b^{-r} = \dfrac{1}{b^{r}}$ : the value stays positive.
A factor $b$ per unit means a factor $b^{\frac12}$ per half-unit.

End of the lesson