What is the expression $\sqrt{8\cdot32\cdot(-3)^2}$ equal to?

Question

What is the expression $\sqrt{8\cdot32\cdot(-3)^2}$ equal to?

Sorry for the basic question. I am a little confused when solving such problems. They are very easy, I know, but still... Which is the easiest algorithm and how would you solve it? We are supposed to use the rule that $$\sqrt{a^2b}=a\sqrt{b},a\ge0, b\ge0,$$ right? What would you do to solve the problem? I was thinking about $$8=4\times2\\32=8\times4\\(-3)^2=3^2$$ but are we supposed to continue this to $$8=4\times2=2\times2\times2=2^3\\32=8\times4=2^3\times2^2=2^5?$$ How can we still use $\sqrt{a^2b}=a\sqrt{b}$ then? Thank you in advance!

Answer 1

It looks like you have the right idea... $8 = 2^3, 32 = 2^5, 8\cdot 32 = 2^8, (-3)^2 = 3^2...$

$\sqrt {2^8\cdot 3^2} = (2^8\cdot 3^2)^\frac 12 = 2^4\cdot 3 = 48$

Answer 2

$\sqrt{8\cdot32\cdot(-3)^2} = \sqrt{8\cdot32\cdot9} = 3 \cdot \sqrt{8\cdot32} = 3 \cdot \sqrt{16\cdot16} = 3 \cdot 16 = 48$

Answer 3

Since $(-3)^2=3^2$, the expression simplifies to $$ \sqrt{8 \cdot 32 \cdot3^2} \, , $$ meaning that you can set $a=3$ and $b=8\cdot32=256$.