√(16x) * √(16x^5)

Steps to Solve

1. Rewrite the expressions with exponent notation

The cube root can be expressed as an exponent of $\frac{1}{3}$, and the square root can be expressed as an exponent of $\frac{1}{2}$. Thus, we rewrite the expression:

$$ \sqrt[3]{16x} = (16x)^{\frac{1}{3}} $$

and

$$ \sqrt{16x^5} = (16x^5)^{\frac{1}{2}} $$

2. Apply the exponent rules

Next, we can multiply the two expressions together. This involves adding the exponents of common bases:

$$ (16x)^{\frac{1}{3}} \cdot (16x^5)^{\frac{1}{2}} = (16^{\frac{1}{3}} \cdot x^{\frac{1}{3}}) \cdot (16^{\frac{1}{2}} \cdot x^{\frac{5}{2}}) $$

3. Combine the bases

Now, we combine the bases of the same type:

$$ 16^{\frac{1}{3} + \frac{1}{2}} \cdot x^{\frac{1}{3} + \frac{5}{2}} $$

4. Calculate the new exponents

Calculating $\frac{1}{3} + \frac{1}{2}$ requires a common denominator (which is 6):

$$ \frac{1}{3} = \frac{2}{6} \quad \text{and} \quad \frac{1}{2} = \frac{3}{6} $$

Thus,

$$ \frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6} $$

For $x$,

$$ \frac{1}{3} + \frac{5}{2} $$

Again, using a common denominator (which is 6):

$$ \frac{1}{3} = \frac{2}{6} \quad \text{and} \quad \frac{5}{2} = \frac{15}{6} $$

Hence,

$$ \frac{1}{3} + \frac{5}{2} = \frac{2}{6} + \frac{15}{6} = \frac{17}{6} $$

5. Final expression

Now we can write the final expression:

$$ 16^{\frac{5}{6}} \cdot x^{\frac{17}{6}} $$