Simplify the expression 45g3h6−−−−−−√ by answering each question and checking your answer before moving to the next step.

Question

Step 2: Rewrite g3−−√ using square roots of perfect square factors. (Hint: you will have two radicals in your first answer.) Then simplify the expression.

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g3−−√=
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Answer 1

To rewrite \( g^{3/2} \) using square roots of perfect square factors, we recognize that \( g^3 \) can be expressed as \( g^2 \cdot g \).

Now we can apply the property of square roots, which states that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).

Therefore,

\[ \sqrt{g^3} = \sqrt{g^2 \cdot g} = \sqrt{g^2} \cdot \sqrt{g} = g \cdot \sqrt{g}. \]

Thus, we can conclude that:

\[ \sqrt{g^3} = g\sqrt{g}. \]

Now, to rewrite \( \sqrt{45g^3h^6} \):

1. Use \( \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \).
2. Recognize \( h^6 = (h^3)^2 \), so \( \sqrt{h^6} = h^3 \).

Putting everything together,

\[ \sqrt{45g^3h^6} = \sqrt{45} \cdot \sqrt{g^3} \cdot \sqrt{h^6} = 3\sqrt{5} \cdot g\sqrt{g} \cdot h^3. \]

So the simplified expression becomes:

\[ 3h^3g\sqrt{5g}. \]

Thus, we have:

1. For \( g^{3/2} \): \( g^{3/2} = g\sqrt{g} \).

Let me know if you need further steps or assistance!