Sure, let's solve the expression [tex]\( 2^{\log _{2 \sqrt{2}} 27} \)[/tex] step-by-step.
1. Understand the expression: We are given [tex]\( 2^{\log _{2 \sqrt{2}} 27} \)[/tex]. Our goal is to simplify this expression to a real number.
2. Change the base of the logarithm: The logarithm [tex]\(\log_{2\sqrt{2}} 27\)[/tex] can be converted to a base-2 logarithm using the change-of-base formula:
[tex]\[
\log_{2\sqrt{2}} 27 = \frac{\log_{2} 27}{\log_{2} (2\sqrt{2})}
\][/tex]
3. Simplify the denominator: Let's simplify [tex]\(\log_{2} (2\sqrt{2})\)[/tex]:
[tex]\[
2\sqrt{2} = 2 \cdot 2^{1/2} = 2^{1 + 1/2} = 2^{3/2}
\][/tex]
Thus,
[tex]\[
\log_{2} (2\sqrt{2}) = \log_{2} (2^{3/2}) = \frac{3}{2}
\][/tex]
4. Substitute back into the expression: Now substitute [tex]\(\log_{2} (2\sqrt{2})\)[/tex] back into our change-of-base formula:
[tex]\[
\log_{2\sqrt{2}} 27 = \frac{\log_{2} 27}{3/2} = \frac{2}{3} \log_{2} 27
\][/tex]
5. Simplify the power: Now we need to simplify [tex]\( 2^{\log_{2\sqrt{2}} 27} \)[/tex]:
[tex]\[
2^{\log_{2\sqrt{2}} 27} = 2^{(\frac{2}{3} \log_{2} 27)}
\][/tex]
6. Substitute values: We need the value of [tex]\(\log_{2} 27\)[/tex]. We know it is approximately [tex]\( 4.754887502163469 \)[/tex].
Using this value:
[tex]\[
\frac{2}{3} \log_{2} 27 = \frac{2}{3} \times 4.754887502163469 \approx 3.1699250014423126
\][/tex]
7. Calculate the final power: Raise 2 to this power:
[tex]\[
2^{3.1699250014423126} \approx 9.000000000000002
\][/tex]
Thus, the value of [tex]\( 2^{\log_{2\sqrt{2}} 27} \)[/tex] is approximately [tex]\( 9.000000000000002 \)[/tex].
