Ans: (a) Explanation: When multiplying powers with the same base, we add the exponents using the product rule: \(a^m \times a^n = a^{m+n}\). So \(3^4 \times 3^2 = 3^{4+2} = 3^6\).
Q2: What is the value of \(5^0\)? (a) 0 (b) 5 (c) 1 (d) Undefined
Solution:
Ans: (c) Explanation: The zero exponent rule states that any non-zero number raised to the power of zero equals 1. Therefore, \(5^0 = 1\).
Ans: (b) Explanation: When dividing powers with the same base, we subtract the exponents using the quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\). So \(\frac{7^8}{7^5} = 7^{8-5} = 7^3\).
Ans: (b) Explanation: When raising a power to another power, we multiply the exponents using the power rule: \((a^m)^n = a^{m \times n}\). So \((2^3)^2 = 2^{3 \times 2} = 2^6\).
Ans: (b) Explanation: The square root of a number is the value that, when multiplied by itself, gives the original number. Since \(8 \times 8 = 64\), we have \(\sqrt{64} = 8\).
Q6: Write \(4^{-3}\) as a positive exponent. (a) \(-4^3\) (b) \(\frac{1}{4^3}\) (c) \(\frac{4}{3}\) (d) \(-\frac{1}{4^3}\)
Solution:
Ans: (b) Explanation: The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). Therefore, \(4^{-3} = \frac{1}{4^3}\).
Ans: (a) Explanation: We can separate the radical into two parts: \(\sqrt{16x^4} = \sqrt{16} \times \sqrt{x^4}\). Since \(\sqrt{16} = 4\) and \(\sqrt{x^4} = x^2\), the result is \(4x^2\).
Q8: Which expression is equivalent to \((3x^2)^3\)? (a) \(3x^5\) (b) \(9x^6\) (c) \(27x^6\) (d) \(3x^6\)
Solution:
Ans: (c) Explanation: Using the power of a product rule, we raise each factor to the power: \((3x^2)^3 = 3^3 \times (x^2)^3\). This gives \(27 \times x^6 = 27x^6\).
## Section B: Fill in the Blanks Q9:The rule \(a^m \times a^n = a^{m+n}\) is called the __________ rule for exponents.
Solution:
Ans: product Explanation: The product rule is used when multiplying powers with the same base by adding their exponents.
Q10:Any non-zero number raised to the power of zero equals __________.
Solution:
Ans: 1 Explanation: The zero exponent rule states that \(a^0 = 1\) for any non-zero value of \(a\).
Q11:The expression \(x^{-5}\) can be written with a positive exponent as __________.
Solution:
Ans: \(\frac{1}{x^5}\) Explanation: The negative exponent rule converts \(x^{-n}\) to \(\frac{1}{x^n}\), making the exponent positive.
Q12:The cube root of 27 is __________ because \(3 \times 3 \times 3 = 27\).
Solution:
Ans: 3 Explanation: The cube root of a number is the value that, when multiplied by itself three times, gives the original number. \(\sqrt[3]{27} = 3\).
Q13:Simplify: \(\sqrt{100} = __________\).
Solution:
Ans: 10 Explanation: The square root of 100 is 10 because \(10 \times 10 = 100\).
Q14:When dividing powers with the same base, we __________ the exponents.
Solution:
Ans: subtract Explanation: The quotient rule for exponents states that \(\frac{a^m}{a^n} = a^{m-n}\), which requires subtracting 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.
Solution:
Ans: Solution: The initial population is \(2^3\). After 4 hours of doubling, we multiply by \(2^4\). Using the product rule: \(2^3 \times 2^4 = 2^{3+4} = 2^7\). Final Answer: \(2^7\) bacteria
Q16:A square garden has an area of 144 square feet. What is the length of one side of the garden?
Solution:
Ans: Solution: The area of a square is \(s^2\), where \(s\) is the side length. We need to find \(s\) such that \(s^2 = 144\). Taking the square root: \(s = \sqrt{144} = 12\). Final Answer: 12 feet
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.
Solution:
Ans: Solution: The original speed is \(10^6\) operations per second. The new speed is \(10^3\) times faster: \(10^6 \times 10^3\). Using the product rule: \(10^6 \times 10^3 = 10^{6+3} = 10^9\). Final Answer: \(10^9\) operations per second
Q18:The volume of a cube is 125 cubic centimeters. What is the length of one edge of the cube?
Solution:
Ans: Solution: The volume of a cube is \(s^3\), where \(s\) is the edge length. We need to find \(s\) such that \(s^3 = 125\). Taking the cube root: \(s = \sqrt[3]{125} = 5\). Final Answer: 5 centimeters
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.
Solution:
Ans: Solution: To find the factor of decrease, we divide the original profit by the new profit. \(\frac{8^5}{8^2}\) Using the quotient rule: \(\frac{8^5}{8^2} = 8^{5-2} = 8^3\). Final Answer: \(8^3\) (or 512 times)
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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