Solving the Square Root Equation

The problem asks us to find the value of the unknown number, represented by '?', in the given equation:

$$ \sqrt{\frac{?}{196}}=\frac{72}{56} $$

Our goal is to isolate '?' and find its numerical value.

Simplifying the Equation

First, let's simplify the fraction on the right side of the equation. Both the numerator (72) and the denominator (56) are divisible by 8.

• $$ \frac{72}{56} = \frac{72 \div 8}{56 \div 8} = \frac{9}{7} $$

Now the equation looks like this:

$$ \sqrt{\frac{?}{196}} = \frac{9}{7} $$

Eliminating the Square Root

To remove the square root on the left side, we need to square both sides of the equation.

• $$ \left(\sqrt{\frac{?}{196}}\right)^2 = \left(\frac{9}{7}\right)^2 $$
• $$ \frac{?}{196} = \frac{9^2}{7^2} $$
• $$ \frac{?}{196} = \frac{81}{49} $$

Calculating the Unknown Value

Now, we need to solve for '?'. We can do this by multiplying both sides of the equation by 196.

• $$ ? = 196 \times \frac{81}{49} $$

We can simplify this calculation by noticing that 196 is a multiple of 49. Let's find out how many times 49 goes into 196:

• $$ 196 \div 49 = 4 $$
• So, we can rewrite 196 as \(4 \times 49\).

Substitute this back into the equation for '?':

• $$ ? = (4 \times 49) \times \frac{81}{49} $$

The '49' in the numerator and denominator cancel out:

• $$ ? = 4 \times 81 $$

Finally, perform the multiplication:

• $$ ? = 324 $$

Conclusion

The value that satisfies the equation is 324. This matches the third option provided.