Sure, let's simplify the expression [tex]\(\sqrt{25 x^5} y 3\)[/tex].
1. Understand and identify each part of the expression:
The given expression is [tex]\(\sqrt{25 x^5} y 3\)[/tex]. This is a product of a square root term, a variable [tex]\(y\)[/tex], and a constant 3.
2. Simplify inside the square root:
First, simplify the term inside the square root: [tex]\(\sqrt{25 x^5}\)[/tex].
- The number 25 is a perfect square, so [tex]\(\sqrt{25} = 5\)[/tex].
- The [tex]\(x^5\)[/tex] can be split for simplification as [tex]\(\sqrt{x^5} = \sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x}\)[/tex].
3. Simplify the square root of [tex]\(x^4\)[/tex]:
- Since [tex]\(\sqrt{x^4} = x^2\)[/tex], we now have [tex]\(\sqrt{x^5} = x^2 \cdot \sqrt{x}\)[/tex].
4. Combine these simplifications:
- Combine the results to get [tex]\(\sqrt{25 x^5} = 5x^2 \sqrt{x}\)[/tex].
5. Substitute back into the original expression:
- Replace [tex]\(\sqrt{25 x^5}\)[/tex] with [tex]\(5x^2 \sqrt{x}\)[/tex], giving us [tex]\(5x^2 \sqrt{x} \cdot y \cdot 3\)[/tex].
6. Simplify the constants:
- Combine the constants [tex]\(5\)[/tex] and [tex]\(3\)[/tex]: [tex]\(5 \cdot 3 = 15\)[/tex].
7. Construct the final simplified form:
- Putting it all together, we get [tex]\(15x^2 \sqrt{x} \cdot y\)[/tex].
8. Reorganize the expression for clarity:
- We typically write such expressions in a more readable form: [tex]\(15 y x^2 \sqrt{x}\)[/tex].
So, the simplified form of the expression [tex]\(\sqrt{25 x^5} y 3\)[/tex] is:
[tex]\[ 15 y \sqrt{x^5} \][/tex]
