Question

Simplify.
Remove all perfect squares from inside the square root. Assume
\[b\] is positive.
\[\sqrt{48b^7}=\]

Answer 1

To simplify the expression \(\sqrt{48b^7}\), we first need to factor 48 and \(b^7\).

1. Factor 48:
\[
48 = 16 \times 3 = 4^2 \times 3
\]

2. For \(b^7\), we can break it down based on perfect squares:
\[
b^7 = b^6 \times b = (b^3)^2 \times b
\]

Now we can combine these factorizations:
\[
48b^7 = 16 \times 3 \times b^6 \times b = (4^2) \times 3 \times (b^3)^2 \times b
\]

3. Now we can simplify the square root:
\[
\sqrt{48b^7} = \sqrt{(4^2) \times 3 \times (b^3)^2 \times b}
\]

4. Using the property \(\sqrt{xy} = \sqrt{x} \sqrt{y}\), we can separate the perfect squares from the square root:
\[
\sqrt{48b^7} = \sqrt{(4^2)} \times \sqrt{3} \times \sqrt{(b^3)^2} \times \sqrt{b}
\]

5. Now compute the square roots of the perfect squares:
\[
= 4 \times \sqrt{3} \times b^3 \times \sqrt{b}
\]

6. Combine the terms:
\[
= 4b^3\sqrt{3b}
\]

Thus, the simplified result is:
\[
\boxed{4b^3\sqrt{3b}}
\]