Solving the Square Root Equation for the Value of x

This solution explains how to find the value of 'x' in the given algebraic equation involving a square root.

Understanding the Equation

We are given the equation:

$\sqrt{\left(1+\frac{27}{169}\right)} = \left(1+\frac{x}{13}\right)$

Our goal is to isolate 'x' and determine its numerical value.

Step-by-Step Solution

1. Simplify the expression inside the square root:

   First, find a common denominator for the terms inside the square root on the left side:

   $1+\frac{27}{169} = \frac{169}{169} + \frac{27}{169} = \frac{169 + 27}{169}$

   Add the numbers in the numerator:

   $\frac{169 + 27}{169} = \frac{196}{169}$

   So, the left side becomes:

   $\sqrt{\left(\frac{196}{169}\right)}$

2. Calculate the square root:

   Find the square root of the numerator and the denominator:

   $\sqrt{\frac{196}{169}} = \frac{\sqrt{196}}{\sqrt{169}}$

   We know that $14 \times 14 = 196$ and $13 \times 13 = 169$. Therefore:

   $\frac{\sqrt{196}}{\sqrt{169}} = \frac{14}{13}$

3. Equate the simplified left side with the right side:

   Now, we replace the left side of the original equation with its simplified value:

   $\frac{14}{13} = \left(1+\frac{x}{13}\right)$

4. Solve for x:

   Rewrite '1' on the right side with a denominator of 13:

   $\frac{14}{13} = \left(\frac{13}{13}+\frac{x}{13}\right)$

   Combine the terms on the right side:

   $\frac{14}{13} = \frac{13+x}{13}$

   Since the denominators are equal, the numerators must also be equal:

   $14 = 13+x$

   Subtract 13 from both sides to find 'x':

   $14 - 13 = x$

   $1 = x$

Final Answer Verification

Substituting $x=1$ back into the original equation:

Left side: $\sqrt{\left(1+\frac{27}{169}\right)} = \frac{14}{13}$

Right side: $\left(1+\frac{1}{13}\right) = \frac{13}{13} + \frac{1}{13} = \frac{14}{13}$

Since the left side equals the right side, our value $x=1$ is correct.

Conclusion

The value of x that satisfies the equation $\sqrt{\left(1+\frac{27}{169}\right)} = \left(1+\frac{x}{13}\right)$ is 1.