The sum of two numbers is 28 and the sum of their squares is 528. Find the square root of the product of both the numbers.

The correct answer is 8√2.

Let the two numbers be a and b. We are given two conditions:

Condition 1: Sum of the two numbers = 28
a+b=28

Condition 2: Sum of their squares = 528
a^2+b^2=528

We need to find the square root of the product of both numbers, i.e., \sqrt{ab}.

Step 1: Use the algebraic identity for (a + b)²

We know that:

{\left(a+b\right)}^2=a^2+2ab+b^2

Substitute the known values:

{\left(28\right)}^2=528+2ab

784=528+2ab

Step 2: Solve for ab

2ab=784-528

2ab=256

ab=128

Step 3: Find the square root of the product

\sqrt{ab}=\sqrt{128}

Break 128 into factors to simplify:

\sqrt{128}=\sqrt{64\times 2}=\sqrt{64}\times \sqrt{2}=8\sqrt{2}

Therefore, the square root of the product of both numbers is 8√2.