Answer :
Let’s go through the given information and the steps to derive the correct statement.
Given:
[tex]\[ \frac{\sqrt{a}}{\sqrt{b}} = x \][/tex]
Our goal is to manipulate this equation to find a relationship between [tex]\( x \)[/tex] and [tex]\(\frac{a}{b}\)[/tex].
### Step-by-step solution:
1. Simplify the left side:
[tex]\[ \frac{\sqrt{a}}{\sqrt{b}} \][/tex]
Rewriting this equation using properties of square roots, we get:
[tex]\[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \][/tex]
2. Square both sides:
To eliminate the square root, we square both sides of the equation:
[tex]\[ \left(\frac{\sqrt{a}}{\sqrt{b}}\right)^2 = x^2 \][/tex]
3. Simplify the squared expression:
When we square the left side, we essentially get the fraction without the square roots:
[tex]\[ \left(\sqrt{\frac{a}{b}}\right)^2 = \frac{a}{b} \][/tex]
4. Equate the simplified form to [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{a}{b} = x^2 \][/tex]
So, the correct statement based on the given equation is:
[tex]\[ x^2 = \frac{a}{b} \][/tex]
### Conclusion:
The statement that must be true is:
[tex]\[ \boxed{D. \, x^2 = \frac{a}{b}} \][/tex]
Given:
[tex]\[ \frac{\sqrt{a}}{\sqrt{b}} = x \][/tex]
Our goal is to manipulate this equation to find a relationship between [tex]\( x \)[/tex] and [tex]\(\frac{a}{b}\)[/tex].
### Step-by-step solution:
1. Simplify the left side:
[tex]\[ \frac{\sqrt{a}}{\sqrt{b}} \][/tex]
Rewriting this equation using properties of square roots, we get:
[tex]\[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \][/tex]
2. Square both sides:
To eliminate the square root, we square both sides of the equation:
[tex]\[ \left(\frac{\sqrt{a}}{\sqrt{b}}\right)^2 = x^2 \][/tex]
3. Simplify the squared expression:
When we square the left side, we essentially get the fraction without the square roots:
[tex]\[ \left(\sqrt{\frac{a}{b}}\right)^2 = \frac{a}{b} \][/tex]
4. Equate the simplified form to [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{a}{b} = x^2 \][/tex]
So, the correct statement based on the given equation is:
[tex]\[ x^2 = \frac{a}{b} \][/tex]
### Conclusion:
The statement that must be true is:
[tex]\[ \boxed{D. \, x^2 = \frac{a}{b}} \][/tex]
