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Which of the following values is equal to the given value?Choose answer: (A) (B) (C) (D)
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Q. Which of the following values is equal to the given value?Choose answer: (A) (B) (C) (D)
- Simplify First Term: Simplify the first term . We know that is , so we can rewrite the term as . Using the power of a power rule , we get .
- Simplify Second Term: Simplify the second term . We know that is , so we can rewrite the term as . Using the power of a product rule , we get $\left(\frac{\(3\)}{\(2\)}\right)^{\(2\)\times\frac{\(17\)}{\(2\)}} \times \left(\frac{\(1\)}{\(2\)}\right)^{\frac{\(17\)}{\(2\)}} = \left(\frac{\(3\)}{\(2\)}\right)^{\(17\)} \times \left(\frac{\(1\)}{\(2\)}\right)^{\frac{\(17\)}{\(2\)}}.
- Combine Simplified Terms: Combine the simplified terms.\(\newline\)We have \((\frac{3}{2})^{11}\) from Step \(1\) and \((\frac{3}{2})^{17} \times (\frac{1}{2})^{\frac{17}{2}}\) from Step \(2\).\(\newline\)Adding these together, we get \((\frac{3}{2})^{11} + (\frac{3}{2})^{17} \times (\frac{1}{2})^{\frac{17}{2}}\).
- Look for Common Factors: Look for common factors in the terms.\(\newline\)We notice that \((\frac{3}{2})^{11}\) is a common factor in both terms.\(\newline\)We can factor it out to get \((\frac{3}{2})^{11} \times (1 + (\frac{3}{2})^{6} \times (\frac{1}{2})^{\frac{17}{2}})\).
- Simplify Inside Parentheses: Simplify the expression inside the parentheses.\(\newline\)We have \(1 + \left(\frac{3}{2}\right)^{6} \times \left(\frac{1}{2}\right)^{\frac{17}{2}}\).\(\newline\)Since \(\left(\frac{3}{2}\right)^{6} = \frac{3^{6}}{2^{6}}\) and \(\left(\frac{1}{2}\right)^{\frac{17}{2}} = \frac{1}{2^{\frac{17}{2}}}\), we can rewrite the expression as \(1 + \left(\frac{3^{6}}{2^{6}}\right) \times \left(\frac{1}{2^{\frac{17}{2}}}\right)\).
- Combine Terms Inside Parentheses: Combine the terms inside the parentheses.\(\newline\)We get \(1 + (3^6 / 2^{6 + 17/2}) = 1 + (3^6 / 2^{19/2})\).
- Simplify Combined Expression: Simplify the combined expression.\(\newline\)We have \((\frac{3}{2})^{11} \times (1 + (\frac{3^6}{2^{\frac{19}{2}}}))\).\(\newline\)This simplifies to \((\frac{3}{2})^{11} \times (1 + \frac{3^6}{2^{\frac{19}{2}}})\).
- Compare with Options: Compare the simplified expression with the given options.\(\newline\)We need to find an option that matches our simplified expression.\(\newline\)Option (A) \(\frac{3^{\frac{39}{2}}}{2^{\frac{67}{2}}}\) does not match because the exponents do not align with our simplified expression.\(\newline\)Option (B) \(\left(\frac{15}{8}\right)^{14}\) does not match because the base and exponent do not align with our simplified expression.\(\newline\)Option (C) \(\frac{3^{11}+3^{\frac{17}{2}}}{2^{\frac{33}{2}}+2^{17}}\) seems to match our simplified expression if we consider that \(3^{\frac{17}{2}}\) is the same as \(3^6 \cdot 3^{\frac{11}{2}}\) and \(2^{\frac{33}{2}}\) is the same as \(2^{\frac{17}{2}} \cdot 2^{\frac{16}{2}}\), which is \(2^{\frac{17}{2}} \cdot 2^8\).\(\newline\)Option (D) \(\frac{3^{11}\cdot\sqrt{2}+3^8\cdot\sqrt{3}}{2^{17}}\) does not match because the exponents and the presence of square roots do not align with our simplified expression.
- Verify Option (C): Verify that option (C) is correct.\(\newline\)We need to ensure that the terms in option (C) match our simplified expression after simplification.\(\newline\)\((3^{11}+3^{(17/2)})/(2^{(33/2)}+2^{17})\) can be rewritten as \((3^{11}+3^{8}\cdot3^{11/2})/(2^{17/2}\cdot2^{16/2}+2^{17})\).\(\newline\)This simplifies to \((3^{11}(1+3^{8}/2^{17/2}))/(2^{17/2}(1+2^{8}))\).\(\newline\)Since \(1+2^{8}\) is not equal to \(2^{17/2}\), option (C) does not match our simplified expression.
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