Questions: Perform the operations and simplify: (sqrt(5 x^7))/(sqrt(3 x)) * sqrt(15 x^2)

Solution Steps

To solve the given mathematical expressions, we need to simplify each one step by step. Here are the high-level ideas for each:

1. For the expression \(\frac{\sqrt{5 x^{7}}}{\sqrt{3 x}} \cdot \sqrt{15 x^{2}}\):

   • Simplify the square roots in the numerator and denominator.
   • Combine the terms and simplify further.

2. For the expression \(5 x^{4}\):

   • This is already in its simplest form.

3. For the expression \(5 x^{2} \sqrt{x^{6}}\):

   • Simplify the square root term.
   • Multiply the simplified term with \(5 x^{2}\).

4. For the expression \(\sqrt{25 x^{8}}\):

   • Simplify the square root term.

5. For the expression \(25 x^{8}\):

   • This is already in its simplest form.

We start with the expression

\[ \frac{\sqrt{5 x^{7}}}{\sqrt{3 x}} \cdot \sqrt{15 x^{2}}. \]

This can be rewritten as

\[ \frac{\sqrt{5} \cdot \sqrt{x^{7}}}{\sqrt{3} \cdot \sqrt{x}} \cdot \sqrt{15} \cdot \sqrt{x^{2}}. \]

Combining the square roots, we have

\[ \frac{\sqrt{5} \cdot \sqrt{15} \cdot \sqrt{x^{7} \cdot x^{2}}}{\sqrt{3} \cdot \sqrt{x}} = \frac{\sqrt{75} \cdot \sqrt{x^{9}}}{\sqrt{3} \cdot \sqrt{x}}. \]

This simplifies to

\[ \frac{\sqrt{75}}{\sqrt{3}} \cdot \sqrt{x^{8}} = \sqrt{25} \cdot \sqrt{x^{8}} = 5 \cdot x^{4}. \]

The second expression is

\[ 5 x^{4}. \]

This expression is already in its simplest form.

The third expression is

\[ 5 x^{2} \sqrt{x^{6}}. \]

Since \(\sqrt{x^{6}} = x^{3}\), we can rewrite this as

\[ 5 x^{2} \cdot x^{3} = 5 x^{5}. \]

The fourth expression is

\[ \sqrt{25 x^{8}}. \]

This simplifies to

\[ \sqrt{25} \cdot \sqrt{x^{8}} = 5 \cdot x^{4}. \]

The fifth expression is

\[ 25 x^{8}. \]

This expression is already in its simplest form.

Final Answer