2.1 Laws of indices for rational exponents — Practise with help

Work each one first, closed book. Reveal a hint only if you are stuck, the next step to see the first move, and the full working only to check.

Warm-up: remember

Work out $2^5$ and $3^3$ .
Work out $\sqrt{64}$ and $\sqrt[3]{64}$ .
Write $\dfrac{1}{4}$ as a power of $2$ .

Answers

$2^5 = 32$ and $3^3 = 27$ .
$\sqrt{64} = 8$ , and $\sqrt[3]{64} = 4$ because $4^3 = 64$ .
$\dfrac{1}{4} = \dfrac{1}{2^2} = 2^{-2}$ .

If question 3 was hard, revisit the negative-index law before you continue.

Practise with help

1. Work out $64^{\frac{1}{3}}$ and $64^{\frac{2}{3}}$ .

Hint

The denominator $3$ names the cube root. Start with the cube root of $64$ , then use the numerator.

Next step

Write $64 = 4^3$ , so that $\sqrt[3]{64} = 4$ . Then raise $4$ to the numerator.

Full working

The cube root of $64$ is $4$ , because $4^3 = 64$ .

64^{\frac{1}{3}} = 4.

For the second power, raise that root to the power $2$ :

64^{\frac{2}{3}} = \left(64^{\frac{1}{3}}\right)^2 = 4^2 = 16.

2. Work out $100^{-\frac{3}{2}}$ .

Hint

The negative sign first turns the expression upside down. The denominator $2$ then names the square root.

Next step

Write $100^{-\frac{3}{2}} = \dfrac{1}{100^{\frac{3}{2}}}$ and work out $100^{\frac{3}{2}}$ on its own.

Full working

The negative exponent gives a reciprocal:

100^{-\frac{3}{2}} = \frac{1}{100^{\frac{3}{2}}}.

Now $100^{\frac{3}{2}} = \left(\sqrt{100}\right)^3 = 10^3 = 1000$ , so

\frac{1}{100^{\frac{3}{2}}} = \frac{1}{1000}.

3. Simplify $\dfrac{x^{\frac{7}{2}}}{x^{\frac{3}{2}}}$ , where $x > 0$ .

Hint

This is division with the same base, so subtract the exponents.

Next step

Subtract the fraction: $\frac{7}{2} - \frac{3}{2}$ .

Full working

Subtract the exponents:

\frac{x^{\frac{7}{2}}}{x^{\frac{3}{2}}} = x^{\frac{7}{2} - \frac{3}{2}} = x^{\frac{4}{2}} = x^{2}.

4. Simplify $\left(27x^3\right)^{\frac{2}{3}}$ , where $x > 0$ .

Hint

The power $\frac{2}{3}$ applies to the number $27$ and to the letter $x^3$ separately.

Next step

Work out $27^{\frac{2}{3}}$ first, then use the power-of-a-power law on $\left(x^3\right)^{\frac{2}{3}}$ .

Full working

Split the bracket:

\left(27x^3\right)^{\frac{2}{3}} = 27^{\frac{2}{3}} \times \left(x^3\right)^{\frac{2}{3}}.

The cube root of $27$ is $3$ , so $27^{\frac{2}{3}} = 3^2 = 9$ . The power of a power multiplies the exponents:

\left(x^3\right)^{\frac{2}{3}} = x^{3 \times \frac{2}{3}} = x^{2}.

So the expression simplifies to $9x^2$ .

5. Solve $x^{\frac{2}{3}} = 16$ .

Hint

Raise both sides to the reciprocal power $\frac{3}{2}$ . Then think about a negative base.

Next step

Work out $16^{\frac{3}{2}}$ by taking the square root of $16$ and cubing it.

Full working

Raise both sides to the power $\frac{3}{2}$ :

x = 16^{\frac{3}{2}} = \left(\sqrt{16}\right)^3 = 4^3 = 64.

The numerator $2$ is even, so a negative base also works:

(-64)^{\frac{2}{3}} = \left(\sqrt[3]{-64}\right)^2 = (-4)^2 = 16.

So $x = 64$ or $x = -64$ . Both values check in the original equation.

6. The side length $s$ of a cube with volume $V$ is $s = V^{\frac{1}{3}}$ . Work out the side length of a cube whose volume is $\frac{27}{64}$ m $^3$ .

Hint

The power $\frac{1}{3}$ is a cube root. Take the cube root of the numerator and the cube root of the denominator.

Next step

Write $\left(\frac{27}{64}\right)^{\frac{1}{3}}$ as $\dfrac{27^{\frac{1}{3}}}{64^{\frac{1}{3}}}$ , then find each root.

Full working

The cube root of a fraction is the fraction of the cube roots:

s = \left(\frac{27}{64}\right)^{\frac{1}{3}} = \frac{27^{\frac{1}{3}}}{64^{\frac{1}{3}}} = \frac{3}{4}.

The side length is $\frac{3}{4}$ m. Check the answer by cubing it: $\left(\frac{3}{4}\right)^3 = \frac{27}{64}$ , which is the volume we started with.