Let's solve the given expression step by step:
Firstly, we need to work with the mixed fraction inside the square root. The mixed fraction given is [tex]\(16 \frac{3}{4}\)[/tex].
1. Convert the mixed fraction to an improper fraction:
A mixed fraction [tex]\(16 \frac{3}{4}\)[/tex] can be converted to an improper fraction by the following steps:
- Multiply the whole number part by the denominator of the fractional part: [tex]\(16 \times 4 = 64\)[/tex].
- Add the numerator of the fractional part to this result: [tex]\(64 + 3 = 67\)[/tex].
- So, [tex]\(16 \frac{3}{4} = \frac{67}{4}\)[/tex].
Now, as a decimal, this improper fraction [tex]\(\frac{67}{4}\)[/tex] equals [tex]\(16.75\)[/tex].
2. Calculate the square root of 16.75:
The square root of 16.75 is approximately [tex]\(4.092676385936225\)[/tex].
3. Calculate the cube root of 53:
The cube root of 53 is approximately [tex]\(3.756285754221072\)[/tex].
4. Form the final expression and perform the division:
We are given the expression [tex]\(\frac{\sqrt{16 \frac{3}{4}}}{\sqrt[3]{53}}\)[/tex]. We substitute the computed values in:
[tex]\[
\frac{\sqrt{16.75}}{\sqrt[3]{53}} = \frac{4.092676385936225}{3.756285754221072}
\][/tex]
5. Perform the division:
[tex]\[
\frac{4.092676385936225}{3.756285754221072} \approx 1.089554057844811
\][/tex]
So, the value of the expression [tex]\(\frac{\sqrt{16 \frac{3}{4}}}{\sqrt[3]{53}}\)[/tex] is approximately [tex]\(1.089554057844811\)[/tex].
