To simplify the given expression [tex]\(3 \sqrt{49 b^3 d}\)[/tex], we first need to simplify the expression inside the square root.
1. The square root of a product can be written as the product of the square roots:
[tex]\[
\sqrt{49 b^3 d} = \sqrt{49} \cdot \sqrt{b^3} \cdot \sqrt{d}
\][/tex]
2. Next, we simplify each term inside the square root:
- [tex]\(\sqrt{49} = 7\)[/tex] since 49 is a perfect square.
- [tex]\(\sqrt{b^3}\)[/tex] can be written as [tex]\(\sqrt{b^2 \cdot b} = \sqrt{b^2} \cdot \sqrt{b} = b \cdot \sqrt{b}\)[/tex].
- [tex]\(\sqrt{d}\)[/tex] remains as it is.
3. Now, substitute these simplified terms back into the product:
[tex]\[
\sqrt{49 b^3 d} = 7 \cdot b \cdot \sqrt{b} \cdot \sqrt{d}
\][/tex]
4. Combine these terms into a single expression:
[tex]\[
3 \cdot 7 \cdot b \cdot \sqrt{b} \cdot \sqrt{d} = 21 b \cdot \sqrt{b} \cdot \sqrt{d}
\][/tex]
5. Recognize that [tex]\(\sqrt{b} = b^{0.5}\)[/tex] and [tex]\(\sqrt{d} = d^{0.5}\)[/tex]:
[tex]\[
21 b \cdot b^{0.5} \cdot d^{0.5}
\][/tex]
6. Combine the exponents for [tex]\(b\)[/tex]:
- Since [tex]\(b = b^1\)[/tex] and [tex]\(b^{0.5}\)[/tex], their combined exponent is [tex]\(1 + 0.5 = 1.5\)[/tex]:
[tex]\[
21 b^{1.5} d^{0.5}
\][/tex]
Therefore, the simplified form of the expression [tex]\(3 \sqrt{49 b^3 d}\)[/tex] is:
[tex]\[
21b^{1.5}d^{0.5}
\][/tex]
The correct answer to be entered in the box is:
[tex]\[
21b^{1.5}d^{0.5}
\][/tex]
