Rationalize the denominator: (2)/(3sqrt(2)−5)

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for questions 6 and 7.

6. Rationalize the denominator: $\frac{2}{3\sqrt{2}-5}$

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of $3\sqrt{2}-5$ is $3\sqrt{2}+5$ .
$\frac{2}{3\sqrt{2}-5} \times \frac{3\sqrt{2}+5}{3\sqrt{2}+5}$
Multiply the numerators and the denominators.
$\frac{2(3\sqrt{2}+5)}{(3\sqrt{2}-5)(3\sqrt{2}+5)}$
Expand the numerator and use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ for the denominator.
$\frac{2 \cdot 3\sqrt{2} + 2 \cdot 5}{(3\sqrt{2})^2 - (5)^2}$ $\frac{6\sqrt{2} + 10}{9 \cdot 2 - 25}$ $\frac{6\sqrt{2} + 10}{18 - 25}$
Simplify the denominator.
$\frac{6\sqrt{2} + 10}{-7}$
This can also be written by moving the negative sign to the front or distributing it to the numerator.
$\boxed{-\frac{6\sqrt{2} + 10}{7}}$
7. Simplify: $14\sqrt{20} - 8\sqrt{20} - 10\sqrt{5}$
Simplify the surd $\sqrt{20}$ by finding its prime factors.
$\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$
Substitute the simplified surd back into the expression.
$14(2\sqrt{5}) - 8(2\sqrt{5}) - 10\sqrt{5}$
Perform the multiplications.
$28\sqrt{5} - 16\sqrt{5} - 10\sqrt{5}$
Combine the like terms (terms with $\sqrt{5}$ ).
$(28 - 16 - 10)\sqrt{5}$ $(12 - 10)\sqrt{5}$ $\boxed{2\sqrt{5}}$

Rationalize the denominator: (2)/(3sqrt(2)-5)