Laws of exponents Problem set B One question at a time

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

• \((2x^3)^4 \times 5x^{-9} = 16x^{12} \times 5x^{-9}\)
• Multiply the numbers, add the powers of x.
• \(80x^{3}\)

Find the exact value of:

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

   • \(3^{-4} = \frac{1}{3^{4}}\)
   • \(= \frac{1}{81}\)

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

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

Find the exact value of:

1. (a) \(81^{\frac{3}{4}}\).

   • \(81^{\frac{3}{4}} = \left(\sqrt[4]{81}\right)^{3} = 3^{3}\)
   • \(= 27\)

2. (b) \(64^{- \frac{2}{3}}\).

   • \(\text{Negative index: take the reciprocal. } 64^{- \frac{2}{3}} = 1 \div 64^{\frac{2}{3}}\)
   • \(64^{\frac{2}{3}} = \left(\sqrt[3]{64}\right)^{2} = 4^{2}\)
   • \(= \frac{1}{16}\)

3. (c) \(\left(\frac{49}{16}\right)^{- \frac{3}{2}}\).

   • \(\text{Negative index: flip the fraction. Use } \left(\frac{16}{49}\right)^{\frac{3}{2}}\)
   • \(\left(\frac{16}{49}\right)^{\frac{3}{2}} = \left(\sqrt{\frac{16}{49}}\right)^{3} = \left(\frac{4}{7}\right)^{3}\)
   • \(= \frac{64}{343}\)

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

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

   • Write each root as a power of x, then use the index laws.
   • \(\frac{9x^5}{6\sqrt[3]{x}} = \frac{3}{2}x^{\frac{14}{3}}\)

2. (b) \(\frac{6\sqrt{x}}{x^4}\);

   • Write each root as a power of x, then use the index laws.
   • \(\frac{6\sqrt{x}}{x^4} = 6x^{- \frac{7}{2}}\)

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

   • Write each root as a power of x, then use the index laws.
   • \(\sqrt[3]{x} \times \sqrt{x^3} = x^{\frac{11}{6}}\)

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

1. (a) \(\frac{8}{\sqrt[4]{x}} + 5x\sqrt{x}\);

   • Write each term as a power of x.
   • \(5x^{\frac{3}{2}} + 8x^{- \frac{1}{4}}\)

2. (b) \(\frac{8x^3+12\sqrt{x}}{4x^2}\).

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

Solve:

1. (a) \(x^{\frac{3}{2}} = \frac{343}{8}\);

   • \(x = \left(\frac{343}{8}\right)^{\frac{2}{3}}\)
   • \(x = \frac{49}{4}\)

2. (b) \(x^{- \frac{2}{3}} = \frac{1}{9}\);

   • \(x = \left(\frac{1}{9}\right)^{- \frac{3}{2}}\)
   • \(x = 27\)

3. (c) \(\sqrt{x} = 5\sqrt[3]{x}\).

   • \(\frac{x^{1/2}}{x^{1/3}} = 5 \Rightarrow x^{\frac{1}{6}} = 5\)
   • \(x = 5^6 = 15625\)

Solve:

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

   • \(7^{\frac{x}{3}} = 7^{2(x-2)}\)
   • \(\frac{x}{3} = 2x - 4\)
   • \(x = \frac{12}{5}\)

2. (b) \(9^{x+2} = \left(\sqrt{27}\right)^{x+10}\).

   • \(3^{2(x+2)} = 3^{\frac{3(x+10)}{2}}\)
   • \(2(x+2) = \frac{3(x+10)}{2}\)
   • \(x = 22\)

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

• \((4x^3)^2 = 16x^{6}\)
• Divide the numbers and subtract the powers: \(16 \div 8\) and \(6 - (-4) = 10\).
• \(2x^{10}\)

Find the exact value of:

1. (a) \(5^{-3}\);

   • \(5^{-3} = \frac{1}{5^3}\)
   • \(= \frac{1}{125}\)

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

   • \(\left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^{3}\)
   • \(= \frac{8}{27}\)

3. (c) \(7^0 - 2^{-2}\).

   • \(7^0 = 1\) and \(2^{-2} = \frac{1}{4}\)
   • \(1 - \frac{1}{4} = \frac{3}{4}\)
   • \(= \frac{3}{4}\)

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

1. (a) Write 0.000 061 3 in standard form.

   • Move the point 5 places right to make a = 6.13
   • \(0.000\,061\,3=6.13\times10^{-5}\)

2. (b) Find \((3.5\times10^{7})\times(6\times10^{-10})\).

   • \(3.5\times6=21\) and \(10^{7}\times10^{-10}=10^{-3}\)
   • \(21\times10^{-3}=2.1\times10^{-2}\)

3. (c) Find \((8.4\times10^{7})+(3\times10^{8})\).

   • \(8.4\times10^{7}=0.84\times10^{8}\)
   • \(0.84\times10^{8}+3\times10^{8}=3.84\times10^{8}\)

4. (d) Find \(\dfrac{1.2\times10^{-2}}{4\times10^{-9}}\).

   • \(1.2\div4=0.3\) and \(10^{-2}\div10^{-9}=10^{7}\)
   • \(0.3\times10^{7}=3\times10^{6}\)

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;

   • \(54 \times \left(\frac{3}{2}\right)^{-2} = 54 \times \left(\frac{2}{3}\right)^{2}\)
   • \(= 24\)

2. (b) \(d = 6\) m.

   • \(54 \times 6^{-2} = \frac{54}{36}\)
   • \(= \frac{3}{2}\)

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

1. (a) Find R when m = 16.

   • \(3 \times 16^{3/4} = 3 \times \left(\sqrt[4]{16}\right)^3 = 3 \times 2^3 = 24\)

2. (b) Find m when R = 81.

   • \(m^{3/4} = 27\)
   • \(m = 27^{4/3} = \left(\sqrt[3]{27}\right)^4 = 3^4 = 81\)

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;

   • \(M = 5 \times 3^{3} = 5 \times 27\)
   • \(= 135\)

2. (b) 2 days before it was measured.

   • \(M = 5 \times 3^{-2} = 5 \times \frac{1}{9}\)
   • \(= \frac{5}{9}\)

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

   • \(8.5\times10^{11}=0.85\times10^{12}\)
   • \(4.2\times10^{12}+0.85\times10^{12}=5.05\times10^{12}\) bytes

2. (b) How many times more does the first drive store than the second? Give your answer to 3 significant figures.

   • \(\dfrac{4.2\times10^{12}}{8.5\times10^{11}}\)
   • From the calculator: 4.94

Answer both parts.

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

   • \(y = 81x^{2/3}\)
   • \(\sqrt[4]{y} = \left(81x^{\frac{2}{3}}\right)^{\frac{1}{4}}\)
   • \(3x^{\frac{1}{6}}\)

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

   • \(y = x^{\frac{7}{2}}\)
   • \(x = y^{1 \div \frac{7}{2}} = y^{\frac{2}{7}}\)

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

• \(\left(x+3\sqrt{x}\right)^2 = x^2 + 6x\sqrt{x} + 9x\)
• Divide each term by x.
• \(x + 6x^{\frac{1}{2}} + 9\)

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

• \((2x^{-3})^3 = 8x^{-9}\)
• Numerator: \(8 \times 9 = 72\) and \(x^{-9} \times x^7 = x^{-2}\).
• Divide by \(12x^{-5}\): the number is \(72 \div 12\) and the power is \(-2 - (-5) = 3\).
• \(6x^{3}\)