Let's evaluate each of the given expressions step-by-step:
### (a) [tex]\(\sqrt{1 \frac{49}{576}}\)[/tex]
First, convert the mixed number [tex]\(1 \frac{49}{576}\)[/tex] into an improper fraction.
1 + [tex]\(\frac{49}{576}\)[/tex] is the same as:
[tex]\[ 1 + \frac{49}{576} = \frac{576}{576} + \frac{49}{576} = \frac{576 + 49}{576} = \frac{625}{576} \][/tex]
Now, take the square root of [tex]\(\frac{625}{576}\)[/tex]:
[tex]\[
\sqrt{\frac{625}{576}} = \frac{\sqrt{625}}{\sqrt{576}} = \frac{25}{24}
\][/tex]
Thus, the result is approximately [tex]\(1.0416666666666667\)[/tex].
### (b) [tex]\(\sqrt{1 \frac{56}{169}}\)[/tex]
Convert the mixed number [tex]\(1 \frac{56}{169}\)[/tex] into an improper fraction:
[tex]\[ 1 + \frac{56}{169} = \frac{169}{169} + \frac{56}{169} = \frac{169 + 56}{169} = \frac{225}{169} \][/tex]
Now, take the square root of [tex]\(\frac{225}{169}\)[/tex]:
[tex]\[
\sqrt{\frac{225}{169}} = \frac{\sqrt{225}}{\sqrt{169}} = \frac{15}{13}
\][/tex]
Thus, the result is approximately [tex]\(1.1538461538461537\)[/tex].
### (c) [tex]\(\sqrt{2 \frac{1}{4}}\)[/tex]
Convert the mixed number [tex]\(2 \frac{1}{4}\)[/tex] into an improper fraction:
[tex]\[ 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \][/tex]
Now, take the square root of [tex]\(\frac{9}{4}\)[/tex]:
[tex]\[
\sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}
\][/tex]
Thus, the result is [tex]\(1.5\)[/tex].
### (d) [tex]\(\sqrt{\frac{1}{16} + \frac{1}{9}}\)[/tex]
First, find a common denominator for [tex]\(\frac{1}{16}\)[/tex] and [tex]\(\frac{1}{9}\)[/tex]. The least common denominator of 16 and 9 is 144:
[tex]\[
\frac{1}{16} = \frac{9}{144}, \quad \frac{1}{9} = \frac{16}{144}
\][/tex]
Add the fractions:
[tex]\[
\frac{1}{16} + \frac{1}{9} = \frac{9}{144} + \frac{16}{144} = \frac{25}{144}
\][/tex]
Now, take the square root of [tex]\(\frac{25}{144}\)[/tex]:
[tex]\[
\sqrt{\frac{25}{144}} = \frac{\sqrt{25}}{\sqrt{144}} = \frac{5}{12}
\][/tex]
Thus, the result is approximately [tex]\(0.4166666666666667\)[/tex].
In summary, the evaluated results are:
- (a) [tex]\(1.0416666666666667\)[/tex]
- (b) [tex]\(1.1538461538461537\)[/tex]
- (c) [tex]\(1.5\)[/tex]
- (d) [tex]\(0.4166666666666667\)[/tex]
