Laws of exponents Problem set B

Write \((2x^3)^4 \times 5x^{-9}\) in the form \(ax^p\).

Find the exact value of:

1. (a) \(3^{-4}\).
2. (b) \(\left(\frac{2}{5}\right)^{-3}\).

Find the exact value of:

1. (a) \(81^{\frac{3}{4}}\).
2. (b) \(64^{- \frac{2}{3}}\).
3. (c) \(\left(\frac{49}{16}\right)^{- \frac{3}{2}}\).

Write each expression in the form \(ax^p\):

1. (a) \(\frac{9x^5}{6\sqrt[3]{x}}\);
2. (b) \(\frac{6\sqrt{x}}{x^4}\);
3. (c) \(\sqrt[3]{x} \times \sqrt{x^3}\).

Write each expression in the form \(ax^p + bx^q\):

1. (a) \(\frac{8}{\sqrt[4]{x}} + 5x\sqrt{x}\);
2. (b) \(\frac{8x^3+12\sqrt{x}}{4x^2}\).

Solve:

1. (a) \(x^{\frac{3}{2}} = \frac{343}{8}\);
2. (b) \(x^{- \frac{2}{3}} = \frac{1}{9}\);
3. (c) \(\sqrt{x} = 5\sqrt[3]{x}\).

Solve:

1. (a) \(\left(\sqrt[3]{7}\right)^x = 49^{x-2}\);
2. (b) \(9^{x+2} = \left(\sqrt{27}\right)^{x+10}\).

Write \(\frac{(4x^3)^2}{8x^{-4}}\) in the form \(ax^p\).

Find the exact value of:

1. (a) \(5^{-3}\);
2. (b) \(\left(\frac{3}{2}\right)^{-3}\);
3. (c) \(7^0 - 2^{-2}\).

Write each answer in the form \(a\times10^{k}\).

1. (a) Write 0.000 061 3 in standard form.
2. (b) Find \((3.5\times10^{7})\times(6\times10^{-10})\).
3. (c) Find \((8.4\times10^{7})+(3\times10^{8})\).
4. (d) Find \(\dfrac{1.2\times10^{-2}}{4\times10^{-9}}\).

The strength of a Wi-Fi signal is \(S = 54d^{-2}\) units at a distance of d metres from the router. Find the exact strength at a distance of:

1. (a) \(d = \frac{3}{2}\) m;
2. (b) \(d = 6\) m.

A mammal of mass m kg has a metabolic rate of \(R = 3m^{3/4}\) watts.

1. (a) Find R when m = 16.
2. (b) Find m when R = 81.

The area of mould on a wall, in cm², is \(M = 5 \times 3^t\), where t is the time in days after it was first measured. A negative t means before it was first measured. Find the exact area:

1. (a) 3 days after it was measured;
2. (b) 2 days before it was measured.

One drive in a data centre stores \(4.2\times10^{12}\) bytes and a second drive stores \(8.5\times10^{11}\) bytes.

1. (a) Find the total number of bytes stored, in the form \(a\times10^{k}\).
2. (b) How many times more does the first drive store than the second? Give your answer to 3 significant figures.

Answer both parts.

1. (a) Given that \(y = 81\sqrt[3]{x^2}\), write \(\sqrt[4]{y}\) in the form \(ax^k\).
2. (b) Given that \(y = x^3\sqrt{x}\), write x in terms of y.

Write \(\frac{\left(x + 3\sqrt{x}\right)^2}{x}\) in the form \(x^a + bx^c + d\).

Write \(\frac{(2x^{-3})^3 \times 9x^7}{12x^{-5}}\) in the form \(ax^p\).