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Answer :
To simplify the expression [tex]\(\sqrt{\frac{1}{25}}\)[/tex] using the quotient rule for square roots, we can follow these steps:
1. Understand the Quotient Rule for Square Roots: The rule says that the square root of a fraction [tex]\(\frac{a}{b}\)[/tex] is equal to the square root of the numerator divided by the square root of the denominator. In mathematical terms, this is expressed as:
[tex]\[
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
\][/tex]
2. Apply the Rule to the Given Expression: We have the expression [tex]\(\sqrt{\frac{1}{25}}\)[/tex]. Let's apply the quotient rule:
[tex]\[
\sqrt{\frac{1}{25}} = \frac{\sqrt{1}}{\sqrt{25}}
\][/tex]
3. Calculate the Square Roots:
- The square root of 1 ([tex]\(\sqrt{1}\)[/tex]) is 1, because 1 × 1 = 1.
- The square root of 25 ([tex]\(\sqrt{25}\)[/tex]) is 5, because 5 × 5 = 25.
4. Divide the Results: Now divide the square root of the numerator by the square root of the denominator:
[tex]\[
\frac{\sqrt{1}}{\sqrt{25}} = \frac{1}{5}
\][/tex]
So, [tex]\(\sqrt{\frac{1}{25}} = \frac{1}{5}\)[/tex], which is a simplified form of the original expression.
1. Understand the Quotient Rule for Square Roots: The rule says that the square root of a fraction [tex]\(\frac{a}{b}\)[/tex] is equal to the square root of the numerator divided by the square root of the denominator. In mathematical terms, this is expressed as:
[tex]\[
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
\][/tex]
2. Apply the Rule to the Given Expression: We have the expression [tex]\(\sqrt{\frac{1}{25}}\)[/tex]. Let's apply the quotient rule:
[tex]\[
\sqrt{\frac{1}{25}} = \frac{\sqrt{1}}{\sqrt{25}}
\][/tex]
3. Calculate the Square Roots:
- The square root of 1 ([tex]\(\sqrt{1}\)[/tex]) is 1, because 1 × 1 = 1.
- The square root of 25 ([tex]\(\sqrt{25}\)[/tex]) is 5, because 5 × 5 = 25.
4. Divide the Results: Now divide the square root of the numerator by the square root of the denominator:
[tex]\[
\frac{\sqrt{1}}{\sqrt{25}} = \frac{1}{5}
\][/tex]
So, [tex]\(\sqrt{\frac{1}{25}} = \frac{1}{5}\)[/tex], which is a simplified form of the original expression.
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