Practice sheet featuring various problems on powers and exponents, including negative exponents and simplification.

Here are the step-by-step solutions for the problems on the worksheet.

Evaluate the following:

(51) $-(3)^{-2}$
* First, handle the exponent. A negative exponent means you take the reciprocal: $3^{-2} = \frac{1}{3^2}$.
* Calculate the square: $\frac{1}{9}$.
* Apply the negative sign in front: $-\frac{1}{9}$.

(53) $(-7)^{-2}$
* The negative exponent flips the base: $\frac{1}{(-7)^2}$.
* Square the negative number (negative times negative is positive): $(-7) \times (-7) = 49$.
* Result: $\frac{1}{49}$.

(54) $(-11)^{-2}$
* Flip the base due to the negative exponent: $\frac{1}{(-11)^2}$.
* Square the base: $(-11) \times (-11) = 121$.
* Result: $\frac{1}{121}$.

(55) $(\frac{1}{3})^{-4}$
* A negative exponent on a fraction flips the fraction upside down: $(\frac{3}{1})^4$ or just $3^4$.
* Calculate $3 \times 3 \times 3 \times 3 = 81$.

(56) $(\frac{1}{2})^{-4}$
* Flip the fraction: $(\frac{2}{1})^4$ or $2^4$.
* Calculate $2 \times 2 \times 2 \times 2 = 16$.

(57) $(\frac{1}{1})^{-4}$
* Flip the fraction: $(\frac{1}{1})^4$. Since $\frac{1}{1} = 1$, this is just $1^4$.
* $1 \times 1 \times 1 \times 1 = 1$.

(58) $(\frac{0}{3})^{-4}$
* Simplify inside first: $\frac{0}{3} = 0$. So we have $0^{-4}$.
* This equals $\frac{1}{0^4}$. Division by zero is undefined.
* Result: Undefined.

(59) $(\frac{-1}{2})^{-1}$
* Power of $-1$ simply flips the fraction.
* Flip $\frac{-1}{2}$ to get $\frac{2}{-1}$.
* Simplify: $-2$.

(60) $(\frac{-1}{5})^{-1}$
* Flip the fraction: $\frac{5}{-1}$.
* Simplify: $-5$.

(61) $(\frac{-1}{7})^{-1}$
* Flip the fraction: $\frac{7}{-1}$.
* Simplify: $-7$.

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Long question: Find the values of the following:

(62) $3^{-1} + 4^{-1}$
* Convert to fractions: $\frac{1}{3} + \frac{1}{4}$.
* Find a common denominator (12): $\frac{4}{12} + \frac{3}{12}$.
* Add numerators: $\frac{7}{12}$.

(63) $(3^0 + 4^{-1}) \times 2^2$
* Inside parentheses: $3^0 = 1$ and $4^{-1} = \frac{1}{4}$. So, $(1 + \frac{1}{4}) = \frac{5}{4}$.
* Outside: $2^2 = 4$.
* Multiply: $\frac{5}{4} \times 4 = 5$.

(64) $(3^{-1} + 4^{-1} + 5^{-1})^0$
* Any non-zero number raised to the power of 0 is 1.
* Since the sum inside is not zero, the answer is 1.

(65) $\left\{ \left[ (\frac{1}{3})^{-1} - (\frac{1}{4})^{-1} \right] \right\}^{-1}$
* Simplify inner brackets first: $(\frac{1}{3})^{-1} = 3$ and $(\frac{1}{4})^{-1} = 4$.
* Subtract: $3 - 4 = -1$.
* Apply outer exponent: $(-1)^{-1}$.
* Flip $-1$ (which is $\frac{-1}{1}$): Result is $\frac{1}{-1} = -1$.

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Simplify the following:

(66) $(4^{-1} \times 3^{-1})^2$
* Combine bases with same exponent: $(4 \times 3)^{-1} = 12^{-1} = \frac{1}{12}$.
* Now square it: $(\frac{1}{12})^2 = \frac{1}{144}$.

(67) $(5^{-1} \div 6^{-1})^3$
* Rewrite division as multiplication by reciprocal: $5^{-1} \times 6^1 = \frac{1}{5} \times 6 = \frac{6}{5}$.
* Cube the result: $(\frac{6}{5})^3 = \frac{216}{125}$.

(68) $(2^{-1} + 3^{-1})^{-1}$
* Inside parentheses: $\frac{1}{2} + \frac{1}{3}$. Common denominator is 6.
* $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.
* Apply outer exponent $-1$ (flip the fraction): $\frac{6}{5}$.

(69) $(3^{-1} \times 4^{-1})^{-1} \times 5^{-1}$
* Inside bracket: $(3 \times 4)^{-1} = 12^{-1}$.
* Apply outer $-1$: $(12^{-1})^{-1} = 12^1 = 12$.
* Multiply by last term: $12 \times 5^{-1} = 12 \times \frac{1}{5} = \frac{12}{5}$.

(70) $(3^2 + 2^2) \times (\frac{1}{2})^3$
* Parentheses 1: $9 + 4 = 13$.
* Parentheses 2: $(\frac{1}{2})^3 = \frac{1}{8}$.
* Multiply: $13 \times \frac{1}{8} = \frac{13}{8}$.

(71) $(3^2 - 2^2) \times (\frac{2}{3})^{-3}$
* Parentheses 1: $9 - 4 = 5$.
* Parentheses 2: Flip fraction and cube: $(\frac{3}{2})^3 = \frac{27}{8}$.
* Multiply: $5 \times \frac{27}{8} = \frac{135}{8}$.

(72) $[(\frac{1}{3})^{-3} - (\frac{1}{2})^{-3}] \div (\frac{1}{4})^{-3}$
* Term 1: $(\frac{3}{1})^3 = 27$.
* Term 2: $(\frac{2}{1})^3 = 8$.
* Bracket: $27 - 8 = 19$.
* Divisor: $(\frac{4}{1})^3 = 64$.
* Divide: $19 \div 64 = \frac{19}{64}$.

(73) $(2^2 + 3^2 - 4^2) \div (\frac{3}{2})^2$
* Numerator part: $4 + 9 - 16 = 13 - 16 = -3$.
* Denominator part: $\frac{9}{4}$.
* Divide: $-3 \div \frac{9}{4} = -3 \times \frac{4}{9}$.
* Simplify: $-\frac{12}{9} = -\frac{4}{3}$.

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Write the following in exponential form:

(74) $(\frac{3}{2})^{-1} \times (\frac{3}{2})^{-1} \times (\frac{3}{2})^{-1} \times (\frac{3}{2})^{-1}$
* When multiplying terms with the same base, add the exponents.
* Base: $\frac{3}{2}$.
* Exponent: $(-1) + (-1) + (-1) + (-1) = -4$.
* Result: $(\frac{3}{2})^{-4}$.

(75) $(\frac{2}{5})^{-2} \times (\frac{2}{5})^{-2} \times (\frac{2}{5})^{-2}$
* Same base, add exponents.
* Exponent: $(-2) + (-2) + (-2) = -6$.
* Result: $(\frac{2}{5})^{-6}$.

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Final Answer:
(51) $-\frac{1}{9}$
(53) $\frac{1}{49}$
(54) $\frac{1}{121}$
(55) $81$
(56) $16$
(57) $1$
(58) Undefined
(59) $-2$
(60) $-5$
(61) $-7$
(62) $\frac{7}{12}$
(63) $5$
(64) $1$
(65) $-1$
(66) $\frac{1}{144}$
(67) $\frac{216}{125}$
(68) $\frac{6}{5}$
(69) $\frac{12}{5}$
(70) $\frac{13}{8}$
(71) $\frac{135}{8}$
(72) $\frac{19}{64}$
(73) $-\frac{4}{3}$
(74) $(\frac{3}{2})^{-4}$
(75) $(\frac{2}{5})^{-6}$