[Math] Express the Radical Simplified form $-\sqrt{27k^7q^8}$

Question

Can someone help me with this problem and explain the steps I need help and I have a test for this type of question. I need to simplify
$$-\sqrt{27k^7q^8}$$

Answer 1

Anything which is a square you can take out of the radical. In particular, $27k^7q^8$ contains a lot of squares:

• $27 = 9 \cdot 3$ ($9$ is a square--it is the square of $3$)

• $k^7 = k^6 \cdot k$ ($k^6$ is a square--it is the square of $k^3$)

• $q^8$ is a square (it is the square of $q^4$)

So you can write $$ 27 k^7 q^8 = (3 \cdot k^3 \cdot q^4)^2 \cdot 3 \cdot k $$ Then $$ -\sqrt{27k^7q^8} = -\sqrt{(3 \cdot k^3 \cdot q^4)^2 \cdot 3 \cdot k} = -3 k^3 q^4 \sqrt{3k}. $$