Laws of exponents Problem set A Worked solutions

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

• \((3x^2)^3 \times 2x^{-5} = 27x^6 \times 2x^{-5}\)
• Multiply the numbers, add the powers of x.
• \(54x\)

Find the exact value of:

1. (a) \(2^{-5}\).

   • \(2^{-5} = \frac{1}{2^{5}}\)
   • \(= \frac{1}{32}\)

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

   • \(\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^{2}\)
   • \(= \frac{16}{9}\)

Find the exact value of:

1. (a) \(16^{\frac{3}{4}}\).

   • \(16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^{3} = 2^{3}\)
   • \(= 8\)

2. (b) \(32^{- \frac{2}{5}}\).

   • \(\text{Negative index: take the reciprocal. } 32^{- \frac{2}{5}} = 1 \div 32^{\frac{2}{5}}\)
   • \(32^{\frac{2}{5}} = \left(\sqrt[5]{32}\right)^{2} = 2^{2}\)
   • \(= \frac{1}{4}\)

3. (c) \(\left(\frac{25}{36}\right)^{- \frac{3}{2}}\).

   • \(\text{Negative index: flip the fraction. Use } \left(\frac{36}{25}\right)^{\frac{3}{2}}\)
   • \(\left(\frac{36}{25}\right)^{\frac{3}{2}} = \left(\sqrt{\frac{36}{25}}\right)^{3} = \left(\frac{6}{5}\right)^{3}\)
   • \(= \frac{216}{125}\)

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

1. (a) \(\frac{10x^4}{4\sqrt{x}}\);

   • Write each root as a power of x, then use the index laws.
   • \(\frac{10x^4}{4\sqrt{x}} = \frac{5}{2}x^{\frac{7}{2}}\)

2. (b) \(\frac{3\sqrt[3]{x}}{x^2}\);

   • Write each root as a power of x, then use the index laws.
   • \(\frac{3\sqrt[3]{x}}{x^2} = 3x^{- \frac{5}{3}}\)

3. (c) \(\sqrt{x} \times \sqrt[3]{x^2}\).

   • Write each root as a power of x, then use the index laws.
   • \(\sqrt{x} \times \sqrt[3]{x^2} = x^{\frac{7}{6}}\)

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

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

   • Write each term as a power of x.
   • \(-4x^{\frac{7}{3}} + 5x^{- \frac{1}{2}}\)

2. (b) \(\frac{9x^2-6\sqrt{x}}{3x^2}\).

   • Divide each term on the top by the bottom, one at a time.
   • \(3 - 2x^{- \frac{3}{2}}\)

Solve:

1. (a) \(x^{\frac{3}{2}} = \frac{125}{27}\);

   • \(x = \left(\frac{125}{27}\right)^{\frac{2}{3}}\)
   • \(x = \frac{25}{9}\)

2. (b) \(x^{- \frac{2}{3}} = \frac{1}{4}\);

   • \(x = \left(\frac{1}{4}\right)^{- \frac{3}{2}}\)
   • \(x = 8\)

3. (c) \(\sqrt{x} = 4\sqrt[3]{x}\).

   • \(\frac{x^{1/2}}{x^{1/3}} = 4 \Rightarrow x^{\frac{1}{6}} = 4\)
   • \(x = 4^6 = 4096\)

Solve:

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

   • \(5^{\frac{x}{3}} = 5^{2(x-4)}\)
   • \(\frac{x}{3} = 2x - 8\)
   • \(x = \frac{24}{5}\)

2. (b) \(4^{x+1} = \left(\sqrt{8}\right)^{x+7}\).

   • \(2^{2(x+1)} = 2^{\frac{3(x+7)}{2}}\)
   • \(2(x+1) = \frac{3(x+7)}{2}\)
   • \(x = 17\)

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

• \((5x^2)^2 = 25x^{4}\)
• Divide the numbers and subtract the powers: \(25 \div 10\) and \(4 - (-3) = 7\).
• \(\frac{5}{2}x^{7}\)

Find the exact value of:

1. (a) \(4^{-3}\);

   • \(4^{-3} = \frac{1}{4^3}\)
   • \(= \frac{1}{64}\)

2. (b) \(\left(\frac{2}{5}\right)^{-3}\);

   • \(\left(\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^{3}\)
   • \(= \frac{125}{8}\)

3. (c) \(5^0 + 3^{-1}\).

   • \(5^0 = 1\) and \(3^{-1} = \frac{1}{3}\)
   • \(1 + \frac{1}{3} = \frac{4}{3}\)
   • \(= \frac{4}{3}\)

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

1. (a) Write 0.000 047 2 in standard form.

   • Move the point 5 places right to make a = 4.72
   • \(0.000\,047\,2=4.72\times10^{-5}\)

2. (b) Find \((4.5\times10^{6})\times(8\times10^{-9})\).

   • \(4.5\times8=36\) and \(10^{6}\times10^{-9}=10^{-3}\)
   • \(36\times10^{-3}=3.6\times10^{-2}\)

3. (c) Find \((7.2\times10^{8})+(5\times10^{9})\).

   • \(7.2\times10^{8}=0.72\times10^{9}\)
   • \(0.72\times10^{9}+5\times10^{9}=5.72\times10^{9}\)

4. (d) Find \(\dfrac{9\times10^{-3}}{1.5\times10^{-8}}\).

   • \(9\div1.5=6\) and \(10^{-3}\div10^{-8}=10^{5}\)
   • \(6\times10^{5}\)

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;

   • \(80 \times \left(\frac{2}{5}\right)^{-2} = 80 \times \left(\frac{5}{2}\right)^{2}\)
   • \(= 500\)

2. (b) \(d = 4\) m.

   • \(80 \times 4^{-2} = \frac{80}{16}\)
   • \(= 5\)

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.

   • \(9^{3/2} = \left(\sqrt{9}\right)^3 = 3^3 = 27\)

2. (b) Find r when T = 125.

   • \(r = 125^{2/3}\)
   • \(= \left(\sqrt[3]{125}\right)^2 = 5^2 = 25\)

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;

   • \(N = 3 \times 2^{4} = 3 \times 16\)
   • \(= 48\)

2. (b) 3 hours before noon.

   • \(N = 3 \times 2^{-3} = 3 \times \frac{1}{8}\)
   • \(= \frac{3}{8}\)

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 \(a\times10^{k}\).

   • \(7.35\times10^{22}=0.0735\times10^{24}\)
   • \(5.97\times10^{24}+0.0735\times10^{24}=6.0435\times10^{24}\)
   • \(6.04\times10^{24}\) kg

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.

   • \(\dfrac{5.97\times10^{24}}{7.35\times10^{22}}\)
   • From the calculator: 81.2

Answer both parts.

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

   • \(y = 125x^{3/2}\)
   • \(\sqrt[3]{y} = \left(125x^{\frac{3}{2}}\right)^{\frac{1}{3}}\)
   • \(5x^{\frac{1}{2}}\)

2. (b) Given that \(y = x^2\sqrt[3]{x}\), write x in terms of y.

   • \(y = x^{\frac{7}{3}}\)
   • \(x = y^{1 \div \frac{7}{3}} = y^{\frac{3}{7}}\)

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

• \(\left(x+2\sqrt{x}\right)^2 = x^2 + 4x\sqrt{x} + 4x\)
• Divide each term by x.
• \(x + 4x^{\frac{1}{2}} + 4\)

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

• \((3x^{-2})^2 = 9x^{-4}\)
• Numerator: \(9 \times 8 = 72\) and \(x^{-4} \times x^3 = x^{-1}\).
• Divide by \(6x^{-4}\): the number is \(72 \div 6\) and the power is \(-1 - (-4) = 3\).
• \(12x^{3}\)