Question

1. simplify: (3√5 − 4√2)²

Explanation:

Step1: Apply the formula \((a - b)^2 = a^2 - 2ab + b^2\)

Here, \(a = 3\sqrt{5}\) and \(b = 4\sqrt{2}\). So we have \((3\sqrt{5})^2 - 2\times(3\sqrt{5})\times(4\sqrt{2})+(4\sqrt{2})^2\)

Step2: Calculate each term

• For \((3\sqrt{5})^2\): Using the property \((c\sqrt{d})^2 = c^2\times d\), we get \(3^2\times5 = 9\times5 = 45\)
• For \(2\times(3\sqrt{5})\times(4\sqrt{2})\): Multiply the coefficients and the square roots separately. \(2\times3\times4 = 24\) and \(\sqrt{5}\times\sqrt{2}=\sqrt{10}\), so this term is \(24\sqrt{10}\)
• For \((4\sqrt{2})^2\): Using the same property as above, \(4^2\times2 = 16\times2 = 32\)

Step3: Combine the terms

Substitute the calculated values back into the expression: \(45 - 24\sqrt{10}+32\)
Then combine the constant terms: \(45 + 32 - 24\sqrt{10}=77 - 24\sqrt{10}\)

Answer:

\(77 - 24\sqrt{10}\)