To divide the given radical expressions [tex]\( \frac{5 \sqrt{x^7}}{\sqrt{x^4}} \)[/tex], let's simplify each part step-by-step:
1. Simplify the Radicals:
- Start with [tex]\( \sqrt{x^7} \)[/tex]. We know that [tex]\( \sqrt{x^7} = (x^7)^{1/2} = x^{7/2} \)[/tex].
- Next, simplify [tex]\( \sqrt{x^4} \)[/tex]. We know that [tex]\( \sqrt{x^4} = (x^4)^{1/2} = x^{4/2} = x^2 \)[/tex].
2. Rewrite the Original Expression:
Using the simplified forms of the radicals, the given expression becomes:
[tex]\[
\frac{5 \sqrt{x^7}}{\sqrt{x^4}} = \frac{5 \cdot x^{7/2}}{x^2}
\][/tex]
3. Simplify the Fraction:
To divide the exponents, use the property of exponents [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[
\frac{5 \cdot x^{7/2}}{x^2} = 5 \cdot x^{(7/2) - 2}
\][/tex]
4. Simplify the Exponents:
Calculate the exponent subtraction:
[tex]\[
5 \cdot x^{(7/2 - 2)} = 5 \cdot x^{(7/2 - 4/2)} = 5 \cdot x^{3/2}
\][/tex]
So, the final simplified expression is:
[tex]\[
\boxed{5 x^{3/2}}
\][/tex]
