Sure, let’s solve the problem step by step.
First, let's identify the given value:
[tex]\[ x = 9 + 4 \sqrt{5} \][/tex]
To proceed with the required expression, we need to determine the square root of [tex]\( x \)[/tex].
1. Calculate the Square Root of [tex]\( x \)[/tex]:
[tex]\[ \sqrt{x} = \sqrt{9 + 4 \sqrt{5}} \][/tex]
After carefully calculating, we find:
[tex]\[ \sqrt{x} ≈ 4.23606797749979 \][/tex]
2. Simplify the Expression [tex]\(\frac{\sqrt{x} - 1}{\sqrt{x}}\)[/tex]:
We need to find the value of:
[tex]\[ \frac{\sqrt{x} - 1}{\sqrt{x}} \][/tex]
3. Substitute the Calculated Value of [tex]\(\sqrt{x}\)[/tex]:
[tex]\[ \sqrt{x} ≈ 4.23606797749979 \][/tex]
So the expression becomes:
[tex]\[ \frac{4.23606797749979 - 1}{4.23606797749979} \][/tex]
4. Simplify the Numerator:
[tex]\[ 4.23606797749979 - 1 = 3.23606797749979 \][/tex]
5. Divide by the Denominator:
[tex]\[ \frac{3.23606797749979}{4.23606797749979} ≈ 0.7639320225002103 \][/tex]
Therefore, the value of [tex]\(\frac{\sqrt{x} - 1}{\sqrt{x}}\)[/tex] is:
[tex]\[ \boxed{0.7639320225002103} \][/tex]
