Sure, let's solve the given expression:
[tex]\[
\frac{16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}}}{\left(16^{\frac{1}{2}}\right)^2}
\][/tex]
First, recall the properties of exponents. When you multiply powers with the same base, you add the exponents:
[tex]\[
16^{\frac{5}{4}} \cdot 16^{\frac{1}{4}} = 16^{\left(\frac{5}{4} + \frac{1}{4}\right)}
\][/tex]
Let's add the exponents in the numerator:
[tex]\[
\frac{5}{4} + \frac{1}{4} = \frac{6}{4} = \frac{3}{2}
\][/tex]
So, the expression in the numerator simplifies to:
[tex]\[
16^{\frac{3}{2}}
\][/tex]
Next, let's simplify the denominator. Recall the property of exponents: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
\left(16^{\frac{1}{2}}\right)^2 = 16^{\left(\frac{1}{2} \cdot 2\right)} = 16^1 = 16
\][/tex]
Thus, the original expression now looks like:
[tex]\[
\frac{16^{\frac{3}{2}}}{16}
\][/tex]
Now, use the property of exponents for division, where you subtract the exponents:
[tex]\[
\frac{16^{\frac{3}{2}}}{16^1} = 16^{\left(\frac{3}{2} - 1\right)}
\][/tex]
Subtract the exponents:
[tex]\[
\frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}
\][/tex]
So, the expression simplifies to:
[tex]\[
16^{\frac{1}{2}}
\][/tex]
Finally, recall that raising a number to the [tex]\(\frac{1}{2}\)[/tex] power is the same as taking the square root:
[tex]\[
16^{\frac{1}{2}} = \sqrt{16} = 4
\][/tex]
Hence, the result of the given expression is:
[tex]\[
4.0
\][/tex]
