Question

Simplify.
Multiply and remove all perfect squares from inside the square roots. Assume
\[z\] is positive.
\[\sqrt{z}\cdot\sqrt{30z^2}\cdot\sqrt{35z^3}=\]

Answer 1

To simplify the expression \(\sqrt{z} \cdot \sqrt{30z^2} \cdot \sqrt{35z^3}\), we can use the property of square roots that states \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Therefore, we can combine the square roots:

\[
\sqrt{z} \cdot \sqrt{30z^2} \cdot \sqrt{35z^3} = \sqrt{z \cdot 30z^2 \cdot 35z^3}
\]

Next, we calculate the product inside the square root:

\[
z \cdot 30z^2 \cdot 35z^3 = 30 \cdot 35 \cdot z^{1 + 2 + 3} = 1050 \cdot z^6
\]

Now substituting this back into the square root gives:

\[
\sqrt{z \cdot 30z^2 \cdot 35z^3} = \sqrt{1050 z^6}
\]

Next, we can separate the square root of the constant and the square root of \(z^6\):

\[
\sqrt{1050 z^6} = \sqrt{1050} \cdot \sqrt{z^6}
\]

Since \(\sqrt{z^6} = z^{6/2} = z^3\), we have:

\[
\sqrt{1050 z^6} = \sqrt{1050} \cdot z^3
\]

Now, we need to simplify \(\sqrt{1050}\) further by removing any perfect squares. We start by factorizing \(1050\):

\[
1050 = 2 \cdot 3 \cdot 5^2 \cdot 7
\]

From this factorization, we can see that \(5^2\) is a perfect square. We can extract this square root:

\[
\sqrt{1050} = \sqrt{2 \cdot 3 \cdot 5^2 \cdot 7} = \sqrt{5^2} \cdot \sqrt{2 \cdot 3 \cdot 7} = 5 \cdot \sqrt{42}
\]

Therefore, substituting back, we now have:

\[
\sqrt{1050 z^6} = 5\sqrt{42} \cdot z^3
\]

Putting it all together, the simplified expression is:

\[
\sqrt{z} \cdot \sqrt{30z^2} \cdot \sqrt{35z^3} = 5z^3\sqrt{42}
\]

Thus, the final simplified answer is:

\[
\boxed{5z^3\sqrt{42}}
\]