Verified Answer :

  • The expression \(\frac{(4.078)^3}{\sqrt{631 \cdot 6}}\) is evaluated step-by-step.
  • The numerator \(4.078^3\) is \(\approx 67.81748\).
  • The denominator \(\sqrt{3786}\) is \(\approx 61.53048\).
  • Dividing the numerator by the denominator yields the final value, \(\boxed{1.102177}\).

Explanation

  1. Problem Analysis

The expression to evaluate is given by \[\frac{(4.078)^3}{\sqrt{631 \cdot 6}}.\] This is a constant numerical expression involving exponentiation, multiplication, and square root operations. Our objective is to compute its numerical value.

  1. Calculating the Numerator

First, compute the value of the numerator, which is \(4.078^3\):
\[4.078^3 = 4.078 \times 4.078 \times 4.078 \approx 67.817482552.\]

  1. Calculating the Denominator

Next, compute the value inside the square root in the denominator, which is \(631 \times 6\):
\[631 \times 6 = 3786.\]
Now, compute the square root of this value:
\[\sqrt{3786} \approx 61.5304802517.\]

  1. Final Calculation

Finally, divide the numerator by the denominator:
\[\frac{67.817482552}{61.5304802517} \approx 1.1021770393.\]
Thus, the value of the expression is approximately \(\boxed{1.102177}\).

Examples

Calculating precise numerical values for complex expressions is essential in fields like structural engineering and physics. For example, if you are determining the scaling factor for a model based on specific physical constants (like the ratio of a volume cubed to the root of a cross-sectional area), you might encounter expressions of this form. Performing these calculations accurately ensures the structural integrity and reliability of the designed model.

Related Questions