Why does $15\sqrt{15} = 15^{3/2}$?

Question

I thought I understood the process of converting a surd to index form, but for the challenge: $15\sqrt{15} $, I don’t understand why $15^{3/2}$ would be the answer (according to the book I’m working from).

I tried a number of Photomath style apps but they didn’t generate the answer $15^{3/2}$, the text I’m working from gives the answer with an explanation and I don’t know anyone in real life who i can ask at this time of the evening so I’ve come here in desperation.

Also, my location (Sydney) is under a very strict lockdown with curfews, so I can’t even go to 7/11 where I presume they sell answers to these kind of problems.

Answer 1

\begin{align*} 15\times\sqrt{15}&=15^1\times15^{\frac 1 2}\\ &=15^{1+\frac 1 2}\\ &=15^{\frac 2 2+\frac 1 2}\\ &=15^{\frac 3 2} \end{align*}

Note that the second step follows because of the rule, $$a^m\times a^n=a^{m+n}$$

Answer 2

Moments after I posted my question, the answer came to me (why is this often the way? lol)

$15 = 15^{2/2}$ Therefore, I can add $15^{2/2}$ to $15^{1/2}$ to get the answer, $15^{3/2}$

Answer 3

This is because $15=15^1$ and $\sqrt{15}=15^{1/2}$. Using indices laws, we get that $$ 15\times\sqrt{15}=15^1\times15^{1/2}=15^{1+1/2}=15^{2/2+1/2}=15^{3/2} \, . $$

Answer 4

We are using that $$a^n\cdot a^m = a^{n+m}$$

to obtain

$$a\sqrt a= a \cdot a^{\frac12}=a^{1+\frac12}=a^{\frac{2+1}2}=a^{\frac32}$$