To solve the problem of finding the equivalent quotient for [tex]\(\sqrt{72 x^3} \div \sqrt{50 x^2}\)[/tex], we need to simplify the expression step-by-step. Let's start the simplification:
### Step 1: Combine the Radicals
We have:
[tex]\[ \frac{\sqrt{72 x^3}}{\sqrt{50 x^2}} \][/tex]
Using the property of radicals [tex]\(\frac{\sqrt{A}}{\sqrt{B}} = \sqrt{\frac{A}{B}}\)[/tex], we get:
[tex]\[ \sqrt{\frac{72 x^3}{50 x^2}} \][/tex]
### Step 2: Simplify the Fraction Inside the Radical
Next, we simplify the fraction inside the radical:
[tex]\[ \frac{72 x^3}{50 x^2} = \frac{72 \cdot x^3}{50 \cdot x^2} \][/tex]
Since [tex]\(x^2 \neq 0\)[/tex], we can divide both the numerator and the denominator by [tex]\(x^2\)[/tex]:
[tex]\[ \frac{72 \cdot x^3}{50 \cdot x^2} = \frac{72}{50} \cdot \frac{x^3}{x^2} = \frac{72}{50} \cdot x \][/tex]
### Step 3: Simplify the Numerical Fraction
The numerical fraction [tex]\(\frac{72}{50}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[ \frac{72}{50} = \frac{72 \div 2}{50 \div 2} = \frac{36}{25} \][/tex]
### Step 4: Combine the Simplified Numerical Fraction with [tex]\(x\)[/tex]
We now have:
[tex]\[ \frac{36}{25} \cdot x \][/tex]
### Step 5: Apply the Radical
Since the entire expression inside the radical is now simplified:
[tex]\[ \sqrt{\frac{36}{25} \cdot x} = \sqrt{\frac{36x}{25}} \][/tex]
### Step 6: Simplify the Radical Expression
The radical of a quotient can be split into the quotient of radicals:
[tex]\[ \sqrt{\frac{36x}{25}} = \frac{\sqrt{36x}}{\sqrt{25}} \][/tex]
### Step 7: Simplify the Radicals
We know that:
[tex]\[ \sqrt{36x} = \sqrt{36} \cdot \sqrt{x} = 6\sqrt{x} \][/tex]
And:
[tex]\[ \sqrt{25} = 5 \][/tex]
Therefore:
[tex]\[ \frac{\sqrt{36x}}{\sqrt{25}} = \frac{6\sqrt{x}}{5} \][/tex]
### Final Answer
So, the simplified expression is:
[tex]\[ \frac{6\sqrt{x}}{5} \][/tex]
Matching this with the given choices:
A. [tex]\(\sqrt{72 x^3 - 50 x^2}\)[/tex]
B. [tex]\(\frac{6 x}{5}\)[/tex]
C. [tex]\(\frac{6 \sqrt{x}}{5}\)[/tex] [tex]\(\leftarrow\)[/tex] Correct Answer
D. [tex]\(\sqrt{22 x}\)[/tex]
Thus, the correct choice equivalent to the given expression is:
[tex]\[ \boxed{\frac{6 \sqrt{x}}{5}} \][/tex]
