Question

which expression is equivalent to the following please show your work.

Answer 1

One of the properties of power is the power of a power property:

$(x^a)^b=x^{a\cdot b}$

Then, the given expression is:

$16x^{15}$

The power of a product property is:

$(a\cdot b)^x=a^x\cdot b^x$

If we apply it to the option D we obtain:

$\begin{gathered} (4x^6)^2\cdot x^3=4^2\cdot(x^6)^2\cdot x^3 \\ \text{Now} \\ 4^2=4\cdot4=16 \\ \text{And} \\ (x^6)^2=x^{6\cdot2}=x^{12} \\ \text{Then we have:} \\ (4x^6)^2\cdot x^3=16x^{12}\cdot x^3 \end{gathered}$

Finally, let's apply the product of powers property:

$x^a\cdot x^b=x^{a+b}$

Then, we obtain:

$(4x^6)^2\cdot x^3=16x^{12+3}=16x^{15}$

The answer is option D.