Number 9. Answers in English

Question

ScanSolve AI Tutor · Accepted Answer

Here's how to simplify the expression:

Simplify the numerator $\sqrt{48}$ .
We look for the largest perfect square factor of 48.
$\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$
Rationalize the denominator by multiplying the expression by the conjugate of the denominator.
The conjugate of $\sqrt{5}+\sqrt{3}$ is $\sqrt{5}-\sqrt{3}$ .
$\frac{4\sqrt{3}}{\sqrt{5}+\sqrt{3}} \times \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}$
Multiply the numerators.
$4\sqrt{3}(\sqrt{5}-\sqrt{3}) = 4\sqrt{3}\sqrt{5} - 4\sqrt{3}\sqrt{3}$ $= 4\sqrt{15} - 4(3)$ $= 4\sqrt{15} - 12$
Multiply the denominators using the difference of squares formula $(x+y)(x-y) = x^2 - y^2$ .
$(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2$ $= 5 - 3$ $= 2$
Combine the simplified numerator and denominator.
$\frac{4\sqrt{15} - 12}{2}$
Divide each term in the numerator by the denominator.
$\frac{4\sqrt{15}}{2} - \frac{12}{2}$ $= 2\sqrt{15} - 6$
The expression in the form $a\sqrt{b} + c$ is:
$2\sqrt{15} - 6$

Answer