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Simplify by rationalizing the denominator:
(4)/(sqrt10-sqrt6)
(1 point)

sqrt10+sqrt6
4

4sqrt10+4sqrt6

(4sqrt10-4sqrt6)/(16-2sqrt15)

Simplify by rationalizing the denominator: $\frac{4}{\sqrt{10}-\sqrt{6}}$ $\newline$ ( $1$ point) $\newline$ $\sqrt{10}+\sqrt{6}$ $\newline$ $4$ $\newline$ $4 \sqrt{10}+4 \sqrt{6}$ $\newline$ $\frac{4 \sqrt{10}-4 \sqrt{6}}{16-2 \sqrt{15}}$

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Simplify radical expressions using conjugates

Full solution

Q. Simplify by rationalizing the denominator:

\frac{4}{\sqrt{10}-\sqrt{6}}

\newline

(

1

point)

\newline

\sqrt{10}+\sqrt{6}

\newline

4

\newline

4 \sqrt{10}+4 \sqrt{6}

\newline

\frac{4 \sqrt{10}-4 \sqrt{6}}{16-2 \sqrt{15}}

Identify Conjugate: Identify the conjugate of the denominator. $\newline$ The conjugate of a binomial $a - b$ is $a + b$ . Therefore, the conjugate of $\sqrt{10} - \sqrt{6}$ is $\sqrt{10} + \sqrt{6}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply the fraction by a form of $1$ that consists of the conjugate of the denominator over itself. $\newline$ $\frac{4}{\sqrt{10}-\sqrt{6}} \times \frac{\sqrt{10}+\sqrt{6}}{\sqrt{10}+\sqrt{6}}$
Apply Distributive Property: Apply the distributive property to the numerator. $\newline$ Multiply $4$ by each term in the conjugate. $\newline$ $4 \times \sqrt{10} + 4 \times \sqrt{6}$ $\newline$ = $4\sqrt{10} + 4\sqrt{6}$
Apply Difference of Squares: Apply the difference of squares formula to the denominator. $\newline$ (\sqrt{\(10\)} + \sqrt{\(6\)})(\sqrt{\(10\)} - \sqrt{\(6\)}) = (\sqrt{\(10\)})^\(2 - (\sqrt{ $6$ })^ $2$ $\newline$ = $10$ - $6$ $\newline$ = $4$
Write Simplified Expression: Write the simplified expression. $\newline$ The numerator is $4\sqrt{10} + 4\sqrt{6}$ and the denominator is $4$ . $\newline$ $(4\sqrt{10} + 4\sqrt{6}) / 4$
Simplify Expression: Simplify the expression by dividing each term in the numerator by the denominator. $\newline$ $(4\sqrt{10} / 4) + (4\sqrt{6} / 4)$ $\newline$ = $\sqrt{10} + \sqrt{6}$

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