$\left(\frac{9}{8}\right)^{\frac{11}{2}}+\left(\frac{3}{4}\right)^{\frac{17}{2}}$ $\newline$ Which of the following values is equal to the given value? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{3^{\frac{39}{2}}}{2^{\frac{67}{2}}}$ $\newline$ (B) $\left(\frac{15}{8}\right)^{14}$ $\newline$ (C) $\frac{3^{11}+3^{\frac{17}{2}}}{2^{\frac{33}{2}}+2^{17}}$ $\newline$ (D) $\frac{3^{11} \cdot \sqrt{2}+3^{8} \cdot \sqrt{3}}{2^{17}}$

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Algebra 1

Compare linear and exponential growth

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\left(\frac{9}{8}\right)^{\frac{11}{2}}+\left(\frac{3}{4}\right)^{\frac{17}{2}}

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Which of the following values is equal to the given value?

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Choose

1

answer:

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(A)

\frac{3^{\frac{39}{2}}}{2^{\frac{67}{2}}}

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(B)

\left(\frac{15}{8}\right)^{14}

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(C)

\frac{3^{11}+3^{\frac{17}{2}}}{2^{\frac{33}{2}}+2^{17}}

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(D)

\frac{3^{11} \cdot \sqrt{2}+3^{8} \cdot \sqrt{3}}{2^{17}}

Rephrasing the problem: First, let's rephrase the ＂Which option is equivalent to the expression $\left(\frac{9}{8}\right)^{\frac{11}{2}}+\left(\frac{3}{4}\right)^{\frac{17}{2}}$ ?＂
Analyzing the expression and answer choices: Now, let's analyze the given expression and the answer choices. We have the expression $\left(\frac{9}{8}\right)^{\frac{11}{2}}+\left(\frac{3}{4}\right)^{\frac{17}{2}}$ . We can simplify this by recognizing that $9 = 3^2$ and $3 = 3^1$ , and similarly for the denominators where $8 = 2^3$ and $4 = 2^2$ .
Simplifying the expression: Rewrite the expression using the base numbers $3$ and $2$ : $\left(\frac{3^2}{2^3}\right)^{\frac{11}{2}} + \left(\frac{3^1}{2^2}\right)^{\frac{17}{2}}$ . Now we can apply the exponent to both the numerator and the denominator separately.
Applying the exponent to the first term: Applying the exponent to the first term: $\left(\left(3^2\right)^{\left(\frac{11}{2}\right)}\right)/\left(\left(2^3\right)^{\left(\frac{11}{2}\right)}\right) = \left(3^{2*\left(\frac{11}{2}\right)}\right)/\left(2^{3*\left(\frac{11}{2}\right)}\right) = 3^{11}/2^{\frac{33}{2}}$ .
Applying the exponent to the second term: Applying the exponent to the second term: $\left(3^1\right)^{\left(\frac{17}{2}\right)}/\left(2^2\right)^{\left(\frac{17}{2}\right)} = 3^{1*\left(\frac{17}{2}\right)}/2^{2*\left(\frac{17}{2}\right)} = 3^{\frac{17}{2}}/2^{17}$ .
Combining the two terms: Combine the two terms: $(3^{11}/2^{33/2}) + (3^{17/2}/2^{17})$ . To add these fractions, we need a common denominator, which would be $2^{17}$ .
Getting a common denominator: To get a common denominator for the first term, multiply both the numerator and the denominator by $2^{(17 - \frac{33}{2})} = 2^{\frac{1}{2}} = \sqrt{2}$ : $(3^{11} \cdot \sqrt{2}) / 2^{17}$ .
Comparing with answer choices: Now we have a common denominator for both terms: $(3^{11} \cdot \sqrt{2} + 3^{\frac{17}{2}}) / 2^{17}$ .
Correcting the discrepancy: Let's compare this result with the answer choices. We can see that choice (D) matches our result: $(3^{11}\sqrt{2}+3^{8}\sqrt{3})/(2^{17})$ . However, there is a slight discrepancy with the second term in the numerator. We have $3^{17/2}$ , which is not the same as $3^8 \cdot \sqrt{3}$ . We need to correct this.