To simplify the expression [tex]\(\sqrt{14 p^3} q \cdot \sqrt{3 p^4}\)[/tex], follow these steps:
1. Combine the expressions under the square roots:
[tex]\[
\sqrt{14 p^3} \cdot \sqrt{3 p^4} = \sqrt{(14 p^3) \cdot (3 p^4)}
\][/tex]
2. Multiply the terms inside the square roots:
[tex]\[
\sqrt{14 \cdot 3 \cdot p^3 \cdot p^4} = \sqrt{42 p^{3+4}} = \sqrt{42 p^7}
\][/tex]
3. Use the properties of square roots to separate the terms involving the variable and the constant factors:
[tex]\[
\sqrt{42 p^7}
\][/tex]
The properties of exponents and square roots allow us to break this down further. Recall that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
[tex]\[
\sqrt{42 p^7} = \sqrt{42} \cdot \sqrt{p^7}
\][/tex]
4. Simplify [tex]\(\sqrt{p^7}\)[/tex]:
We can write [tex]\(p^7\)[/tex] as [tex]\(p^6 \cdot p\)[/tex] to apply the square root.
[tex]\[
\sqrt{p^7} = \sqrt{p^6 \cdot p} = \sqrt{p^6} \cdot \sqrt{p} = p^3 \cdot \sqrt{p}
\][/tex]
5. Combine all parts together:
Putting the simplified forms together:
[tex]\[
\sqrt{42} \cdot \sqrt{p^7} = \sqrt{42} \cdot (p^3 \cdot \sqrt{p})
\][/tex]
6. Include the [tex]\(q\)[/tex] term:
The original expression includes a [tex]\(q\)[/tex] multiplicative factor:
[tex]\[
\sqrt{42} \cdot p^3 \cdot \sqrt{p} \cdot q
\][/tex]
This leads us to the final simplified expression:
[tex]\[
\sqrt{42} \cdot q \cdot \sqrt{p^3} \cdot \sqrt{p^4}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\sqrt{42} \cdot q \cdot \sqrt{p^3} \cdot \sqrt{p^4}
\][/tex]
This is the detailed, simplified expression for [tex]\(\sqrt{14 p^3} q \cdot \sqrt{3 p^4}\)[/tex].
