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

\]

Step 7 : Final Answer

The final answer is:

\[

8^{\frac{2}{3}}+\log _{2} 32=9

\]