Laws of exponents Problem set A

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

Find the exact value of:

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

Find the exact value of:

1. (a) \(16^{\frac{3}{4}}\).
2. (b) \(32^{- \frac{2}{5}}\).
3. (c) \(\left(\frac{25}{36}\right)^{- \frac{3}{2}}\).

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

1. (a) \(\frac{10x^4}{4\sqrt{x}}\);
2. (b) \(\frac{3\sqrt[3]{x}}{x^2}\);
3. (c) \(\sqrt{x} \times \sqrt[3]{x^2}\).

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

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

Solve:

1. (a) \(x^{\frac{3}{2}} = \frac{125}{27}\);
2. (b) \(x^{- \frac{2}{3}} = \frac{1}{4}\);
3. (c) \(\sqrt{x} = 4\sqrt[3]{x}\).

Solve:

1. (a) \(\left(\sqrt[3]{5}\right)^x = 25^{x-4}\);
2. (b) \(4^{x+1} = \left(\sqrt{8}\right)^{x+7}\).

Write \(\frac{(5x^2)^2}{10x^{-3}}\) in the form \(ax^p\).

Find the exact value of:

1. (a) \(4^{-3}\);
2. (b) \(\left(\frac{2}{5}\right)^{-3}\);
3. (c) \(5^0 + 3^{-1}\).

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

1. (a) Write 0.000 047 2 in standard form.
2. (b) Find \((4.5\times10^{6})\times(8\times10^{-9})\).
3. (c) Find \((7.2\times10^{8})+(5\times10^{9})\).
4. (d) Find \(\dfrac{9\times10^{-3}}{1.5\times10^{-8}}\).

The brightness of a lamp is \(B = 80d^{-2}\) lux at a distance of d metres. Find the exact brightness at a distance of:

1. (a) \(d = \frac{2}{5}\) m;
2. (b) \(d = 4\) m.

A planet orbits the Sun at a distance of r astronomical units. Its orbital time is \(T = r^{3/2}\) years.

1. (a) Find T when r = 9.
2. (b) Find r when T = 125.

The number of bacteria in a dish, in thousands, is \(N = 3 \times 2^t\), where t is the time in hours after noon. A negative t means before noon. Find the exact number of thousands of bacteria:

1. (a) 4 hours after noon;
2. (b) 3 hours before noon.

The mass of the Earth is \(5.97\times10^{24}\) kg and the mass of the Moon is \(7.35\times10^{22}\) kg.

1. (a) Find their total mass, in the form \(a\times10^{k}\).
2. (b) How many times greater is the mass of the Earth than the mass of the Moon? Give your answer to 3 significant figures.

Answer both parts.

1. (a) Given that \(y = 125\sqrt{x^3}\), write \(\sqrt[3]{y}\) in the form \(ax^k\).
2. (b) Given that \(y = x^2\sqrt[3]{x}\), write x in terms of y.

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

Write \(\frac{(3x^{-2})^2 \times 8x^3}{6x^{-4}}\) in the form \(ax^p\).