To simplify the expression [tex]\(\sqrt{\frac{20}{49}}\)[/tex], we will proceed step-by-step:
1. Understand the expression:
We have a square root of a fraction: [tex]\(\sqrt{\frac{20}{49}}\)[/tex].
2. Apply the property of square roots on a fraction:
The square root of a fraction is the fraction of the square roots of the numerator and the denominator. Thus:
[tex]\[
\sqrt{\frac{20}{49}} = \frac{\sqrt{20}}{\sqrt{49}}
\][/tex]
3. Simplify the denominator:
The square root of 49 is a perfect square. We know that:
[tex]\[
\sqrt{49} = 7
\][/tex]
So, the denominator simplifies to 7:
[tex]\[
\frac{\sqrt{20}}{\sqrt{49}} = \frac{\sqrt{20}}{7}
\][/tex]
4. Simplify the numerator:
To further simplify the numerator, we need to find the square root of 20. While [tex]\(\sqrt{20}\)[/tex] is not a perfect square, we can express [tex]\(\sqrt{20}\)[/tex] in its decimal form. The value of [tex]\(\sqrt{20}\)[/tex] is approximately:
[tex]\[
\sqrt{20} \approx 4.47213595499958
\][/tex]
Therefore:
[tex]\[
\frac{\sqrt{20}}{7} = \frac{4.47213595499958}{7}
\][/tex]
5. Combine the results:
Thus, the simplified form of [tex]\(\sqrt{\frac{20}{49}}\)[/tex] in decimal form is:
[tex]\[
\frac{4.47213595499958}{7} \approx 0.6388765649999399
\][/tex]
In conclusion, the simplified numerator is approximately [tex]\(4.47213595499958\)[/tex], the simplified denominator is [tex]\(7\)[/tex], and the simplified form of the expression [tex]\(\sqrt{\frac{20}{49}}\)[/tex] is approximately [tex]\(0.6388765649999399\)[/tex].
