301 Moved Permanently - Free Printable

Problem Set: Power and Exponent

We will solve the problems step by step, categorizing them into sections as given in the image.

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#### Section 1: Evaluate the following

1. (51) \(-3^{-2}\)
\[
-3^{-2} = -\frac{1}{3^2} = -\frac{1}{9}
\]

2. (53) \((-7)^{-2}\)
\[
(-7)^{-2} = \frac{1}{(-7)^2} = \frac{1}{49}
\]

3. (54) \((-11)^{-2}\)
\[
(-11)^{-2} = \frac{1}{(-11)^2} = \frac{1}{121}
\]

4. (55) \(\left(\frac{1}{3}\right)^{-4}\)
\[
\left(\frac{1}{3}\right)^{-4} = \left(3\right)^4 = 81
\]

5. (56) \(\left(\frac{1}{2}\right)^{-4}\)
\[
\left(\frac{1}{2}\right)^{-4} = \left(2\right)^4 = 16
\]

6. (57) \(\left(\frac{1}{1}\right)^{-4}\)
\[
\left(\frac{1}{1}\right)^{-4} = 1^{-4} = 1
\]

7. (58) \(\left(\frac{0}{3}\right)^{-4}\)
\[
\left(\frac{0}{3}\right)^{-4} = 0^{-4}
\]
This is undefined because division by zero is not allowed.

8. (59) \(\left(\frac{-1}{2}\right)^{-1}\)
\[
\left(\frac{-1}{2}\right)^{-1} = \frac{1}{\left(\frac{-1}{2}\right)} = -2
\]

9. (60) \(\left(\frac{-1}{5}\right)^{-1}\)
\[
\left(\frac{-1}{5}\right)^{-1} = \frac{1}{\left(\frac{-1}{5}\right)} = -5
\]

10. (61) \(\left(\frac{-1}{7}\right)^{-1}\)
\[
\left(\frac{-1}{7}\right)^{-1} = \frac{1}{\left(\frac{-1}{7}\right)} = -7
\]

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#### Section 2: Long Question: Find the values of the following

11. (62) \(3^{-1} + 4^{-1}\)
\[
3^{-1} + 4^{-1} = \frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}
\]

12. (63) \((3^0 + 4^{-1}) \times 2^2\)
\[
3^0 + 4^{-1} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}
\]
\[
(3^0 + 4^{-1}) \times 2^2 = \frac{5}{4} \times 4 = 5
\]

13. (64) \((3^{-1} + 4^{-1} + 5^{-1})^0\)
\[
3^{-1} + 4^{-1} + 5^{-1} = \frac{1}{3} + \frac{1}{4} + \frac{1}{5}
\]
\[
\text{Common denominator: } 60
\]
\[
\frac{1}{3} + \frac{1}{4} + \frac{1}{5} = \frac{20}{60} + \frac{15}{60} + \frac{12}{60} = \frac{47}{60}
\]
\[
(3^{-1} + 4^{-1} + 5^{-1})^0 = \left(\frac{47}{60}\right)^0 = 1
\]

14. (65) \(\left\{\left(\frac{1}{3}\right)^{-1} - \left(\frac{1}{4}\right)^{-1}\right\}^{-1}\)
\[
\left(\frac{1}{3}\right)^{-1} = 3, \quad \left(\frac{1}{4}\right)^{-1} = 4
\]
\[
\left(\frac{1}{3}\right)^{-1} - \left(\frac{1}{4}\right)^{-1} = 3 - 4 = -1
\]
\[
\left\{\left(\frac{1}{3}\right)^{-1} - \left(\frac{1}{4}\right)^{-1}\right\}^{-1} = (-1)^{-1} = -1
\]

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#### Section 3: Simplify the following

15. (66) \(\left(4^{-1} \times 3^{-1}\right)^2\)
\[
4^{-1} \times 3^{-1} = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}
\]
\[
\left(4^{-1} \times 3^{-1}\right)^2 = \left(\frac{1}{12}\right)^2 = \frac{1}{144}
\]

16. (67) \(\left(5^{-1} \div 6^{-1}\right)^3\)
\[
5^{-1} \div 6^{-1} = \frac{1}{5} \div \frac{1}{6} = \frac{1}{5} \times 6 = \frac{6}{5}
\]
\[
\left(5^{-1} \div 6^{-1}\right)^3 = \left(\frac{6}{5}\right)^3 = \frac{6^3}{5^3} = \frac{216}{125}
\]

17. (68) \(\left(2^{-1} + 3^{-1}\right)^{-1}\)
\[
2^{-1} + 3^{-1} = \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\]
\[
\left(2^{-1} + 3^{-1}\right)^{-1} = \left(\frac{5}{6}\right)^{-1} = \frac{6}{5}
\]

18. (69) \(\left(3^{-1} \times 4^{-1}\right)^{-1} \times 5^{-1}\)
\[
3^{-1} \times 4^{-1} = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}
\]
\[
\left(3^{-1} \times 4^{-1}\right)^{-1} = \left(\frac{1}{12}\right)^{-1} = 12
\]
\[
\left(3^{-1} \times 4^{-1}\right)^{-1} \times 5^{-1} = 12 \times \frac{1}{5} = \frac{12}{5}
\]

19. (70) \(\left(3^2 + 2^2\right) \times \left(\frac{1}{2}\right)^3\)
\[
3^2 + 2^2 = 9 + 4 = 13
\]
\[
\left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]
\[
\left(3^2 + 2^2\right) \times \left(\frac{1}{2}\right)^3 = 13 \times \frac{1}{8} = \frac{13}{8}
\]

20. (71) \(\left(3^2 - 2^2\right) \times \left(\frac{2}{3}\right)^{-3}\)
\[
3^2 - 2^2 = 9 - 4 = 5
\]
\[
\left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}
\]
\[
\left(3^2 - 2^2\right) \times \left(\frac{2}{3}\right)^{-3} = 5 \times \frac{27}{8} = \frac{135}{8}
\]

21. (72) \(\left[\left(\frac{1}{3}\right)^{-3} - \left(\frac{1}{2}\right)^{-3}\right] \div \left(\frac{1}{4}\right)^{-3}\)
\[
\left(\frac{1}{3}\right)^{-3} = 3^3 = 27, \quad \left(\frac{1}{2}\right)^{-3} = 2^3 = 8
\]
\[
\left(\frac{1}{3}\right)^{-3} - \left(\frac{1}{2}\right)^{-3} = 27 - 8 = 19
\]
\[
\left(\frac{1}{4}\right)^{-3} = 4^3 = 64
\]
\[
\left[\left(\frac{1}{3}\right)^{-3} - \left(\frac{1}{2}\right)^{-3}\right] \div \left(\frac{1}{4}\right)^{-3} = 19 \div 64 = \frac{19}{64}
\]

22. (73) \(\left(2^2 + 3^2 - 4^2\right) \div \left(\frac{3}{2}\right)^2\)
\[
2^2 + 3^2 - 4^2 = 4 + 9 - 16 = -3
\]
\[
\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4}
\]
\[
\left(2^2 + 3^2 - 4^2\right) \div \left(\frac{3}{2}\right)^2 = -3 \div \frac{9}{4} = -3 \times \frac{4}{9} = -\frac{12}{9} = -\frac{4}{3}
\]

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#### Section 4: Write the following in exponential form

23. (74) \(\left(\frac{3}{2}\right)^{-1} \times \left(\frac{3}{2}\right)^{-1} \times \left(\frac{3}{2}\right)^{-1} \times \left(\frac{3}{2}\right)^{-1}\)
\[
\left(\frac{3}{2}\right)^{-1} \times \left(\frac{3}{2}\right)^{-1} \times \left(\frac{3}{2}\right)^{-1} \times \left(\frac{3}{2}\right)^{-1} = \left(\frac{3}{2}\right)^{-1+(-1)+(-1)+(-1)} = \left(\frac{3}{2}\right)^{-4}
\]

24. (75) \(\left(\frac{2}{5}\right)^{-2} \times \left(\frac{2}{5}\right)^{-2} \times \left(\frac{2}{5}\right)^{-2}\)
\[
\left(\frac{2}{5}\right)^{-2} \times \left(\frac{2}{5}\right)^{-2} \times \left(\frac{2}{5}\right)^{-2} = \left(\frac{2}{5}\right)^{-2+(-2)+(-2)} = \left(\frac{2}{5}\right)^{-6}
\]

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Final Answers:

\[
\boxed{
\begin{aligned}
&\text{(51)} -\frac{1}{9}, \quad \text{(53)} \frac{1}{49}, \quad \text{(54)} \frac{1}{121}, \quad \text{(55)} 81, \quad \text{(56)} 16, \quad \text{(57)} 1, \quad \text{(58)} \text{Undefined}, \\
&\text{(59)} -2, \quad \text{(60)} -5, \quad \text{(61)} -7, \quad \text{(62)} \frac{7}{12}, \quad \text{(63)} 5, \quad \text{(64)} 1, \quad \text{(65)} -1, \\
&\text{(66)} \frac{1}{144}, \quad \text{(67)} \frac{216}{125}, \quad \text{(68)} \frac{6}{5}, \quad \text{(69)} \frac{12}{5}, \quad \text{(70)} \frac{13}{8}, \quad \text{(71)} \frac{135}{8}, \\
&\text{(72)} \frac{19}{64}, \quad \text{(73)} -\frac{4}{3}, \quad \text{(74)} \left(\frac{3}{2}\right)^{-4}, \quad \text{(75)} \left(\frac{2}{5}\right)^{-6}.
\end{aligned}
}
\]