Calculate (2)/(3-sqrt(2)) and sqrt(25/4)

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems you sent.

First, let's simplify $\sqrt{\frac{25}{4}}$ .

Apply the property of square roots for fractions, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ .
$\sqrt{\frac{25}{4}} = \frac{\sqrt{25}}{\sqrt{4}}$
Calculate the square roots of the numerator and the denominator.
$\sqrt{25} = 5$ $\sqrt{4} = 2$
Substitute the values back into the fraction.
$\frac{5}{2}$
The simplified form is:
$\boxed{\frac{5}{2}}$
Next, let's rationalize the denominator of $\frac{2}{3+\sqrt{2}}$ .
Identify the conjugate of the denominator.
The denominator is $3+\sqrt{2}$ , so its conjugate is $3-\sqrt{2}$ .
Multiply both the numerator and the denominator by the conjugate.
$\frac{2}{3+\sqrt{2}} \times \frac{3-\sqrt{2}}{3-\sqrt{2}}$
Multiply the numerators.
$2(3-\sqrt{2}) = 6 - 2\sqrt{2}$
Multiply the denominators using the difference of squares formula, $(a+b)(a-b) = a^2 - b^2$ .
$(3+\sqrt{2})(3-\sqrt{2}) = 3^2 - (\sqrt{2})^2$ $= 9 - 2$ $= 7$
Combine the simplified numerator and denominator.
$\frac{6 - 2\sqrt{2}}{7}$
The rationalized expression is:
$\boxed{\frac{6 - 2\sqrt{2}}{7}}$

Calculate (2)/(3-sqrt(2)) and sqrt((25)/(4))