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Worksheet (with Solutions): Exponents & Radicals

# Worksheet: Exponents & Radicals ## Section A: Multiple Choice Questions

Q1: Simplify the expression \(3^4 \times 3^2\).
(a) \(3^6\)
(b) \(3^8\)
(c) \(9^6\)
(d) \(6^6\)

Q2: What is the value of \(5^0\)?
(a) 0
(b) 5
(c) 1
(d) Undefined

Q3: Simplify \(\frac{7^8}{7^5}\).
(a) \(7^{13}\)
(b) \(7^3\)
(c) \(7^{40}\)
(d) \(1^3\)

Q4: Evaluate \((2^3)^2\).
(a) \(2^5\)
(b) \(2^6\)
(c) \(2^9\)
(d) \(4^6\)

Q5: Simplify \(\sqrt{64}\).
(a) 4
(b) 8
(c) 32
(d) 16

Q6: Write \(4^{-3}\) as a positive exponent.
(a) \(-4^3\)
(b) \(\frac{1}{4^3}\)
(c) \(\frac{4}{3}\)
(d) \(-\frac{1}{4^3}\)

Q7: Simplify \(\sqrt{16x^4}\).
(a) \(4x^2\)
(b) \(8x^2\)
(c) \(4x^4\)
(d) \(16x^2\)

Q8: Which expression is equivalent to \((3x^2)^3\)?
(a) \(3x^5\)
(b) \(9x^6\)
(c) \(27x^6\)
(d) \(3x^6\)

## Section B: Fill in the Blanks Q9: The rule \(a^m \times a^n = a^{m+n}\) is called the __________ rule for exponents.
Q10: Any non-zero number raised to the power of zero equals __________.
Q11: The expression \(x^{-5}\) can be written with a positive exponent as __________.
Q12: The cube root of 27 is __________ because \(3 \times 3 \times 3 = 27\).
Q13: Simplify: \(\sqrt{100} = __________\).
Q14: When dividing powers with the same base, we __________ the exponents.
## Section C: Word Problems Q15: A bacteria colony doubles every hour. If the initial population is represented by \(2^3\) bacteria, what will the population be after 4 more hours? Express your answer using exponential notation.
Q16: A square garden has an area of 144 square feet. What is the length of one side of the garden?
Q17: A computer's processing speed is measured as \(10^6\) operations per second. If a new model is \(10^3\) times faster, what is the new processing speed? Express your answer in exponential form.
Q18: The volume of a cube is 125 cubic centimeters. What is the length of one edge of the cube?
Q19: A company's profits decreased from \(\$8^5\) to \(\$8^2\) over a period of time. By what factor did the profits decrease? Express your answer in exponential form.
Q20: A rectangular storage box has a square base with side length \(\sqrt{36}\) inches and a height of \(\sqrt{49}\) inches. What is the volume of the box?
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