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Simplify, without use of a calculator:
$8^{\frac{2}{3}}+\log _{2} 32$
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Step 1 : Try to write any quantities as exponents 8 can be written as $2^{3} .32$ can be written as $2^{5}$.
Step 2: Re-write the question using the exponential forms of the numbers
$8^{\frac{2}{3}}+\log _{2} 32=\left(2^{3}\right)^{\frac{2}{3}}+\log _{2} 2^{5}$
Step 3 : Determine which laws can be used.
We can use:
$\log _{a}\left(x^{b}\right)=b \log _{a}(x)$
Step 4 : Apply log laws to simplify
$\left(2^{3}\right)^{\frac{2}{3}}+\log _{2} 2^{5}=(2)^{3 \times \frac{2}{3}}+5 \log _{2} 2$
Step 5 : Determine which laws can be used.
We can now use $\log _{a} a=1$
Step 6 : Apply log laws to simplify
$(2)^{2}+5 \log _{2} 2=2^{2}+5(1)=4+5=9$
$8^{\frac{2}{3}}+\log _{2} 32=9$
by Diamond (66,887 points)

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