square root of 528

Steps to Solve

1. Identify the square root operation We need to find a number $x$ such that $x^2 = 528$. This means we are looking for $x = \sqrt{528}$.

2. Use a calculator or simplify the expression To find the value of $x$, we can either use a calculator or simplify the number first, if possible. First, we can break down 528 into its prime factors to ease the square root calculation.

3. Factor 528 Start with division:

• $528$ is even, divide by $2$: $528 \div 2 = 264$
• $264$ is even, divide by $2$: $264 \div 2 = 132$
• $132$ is even, divide by $2$: $132 \div 2 = 66$
• $66$ is even, divide by $2$: $66 \div 2 = 33$
• $33$ is divisible by $3$: $33 \div 3 = 11$

Thus, the prime factorization of 528 is: $$ 528 = 2^4 \times 3^1 \times 11^1 $$

4. Calculate the square root using properties of exponents Using the prime factorization result, we can take the square root:

• For $2^4$: The square root is $2^{4/2} = 2^2 = 4$.
• For $3^1$: The square root is $3^{1/2} = \sqrt{3}$.
• For $11^1$: The square root is $11^{1/2} = \sqrt{11}$.

Combine these: $$ \sqrt{528} = 4 \cdot \sqrt{3 \cdot 11} = 4 \cdot \sqrt{33} $$

5. Use a calculator for a numerical answer Now we can calculate an approximate value using a calculator: $$ \sqrt{528} \approx 22.976 $$