Write the expression in simplest form. (9)/(sqrt3+sqrt7)=(◻)/(◻)

Rationalize the Denominator: To simplify the expression (9)/(sqrt3+sqrt7), we need to rationalize the denominator. This means we need to eliminate the square roots in the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of (sqrt3+sqrt7) is (sqrt3-sqrt7).Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator: (9)/(sqrt3+sqrt7) * (sqrt3-sqrt7)/(sqrt3-sqrt7).Apply Distributive Property: Use the distributive property (a+b)(a-b) = a^2 - b^2 to multiply the denominators: (sqrt3+sqrt7)(sqrt3-sqrt7) = (sqrt3)^2 - (sqrt7)^2.Calculate Squares: Calculate the squares of the square roots: (sqrt3)^2 - (sqrt7)^2 = 3 - 7.Subtract Numbers: Subtract the numbers to find the value of the denominator: 3 - 7 = -4.Multiply Numerator: Now, multiply the numerator by the conjugate of the denominator: 9 * (sqrt3-sqrt7).Divide by Denominator: Since the denominator is a constant (-4), we can divide the terms in the numerator by the denominator: (9 * sqrt3)/(-4) - (9 * sqrt7)/(-4).Simplify Fractions: Simplify the fractions by dividing 9 by -4: (-9/4) * sqrt3 - (-9/4) * sqrt7.Write Final Expression: Write the final simplified expression: (9)/(sqrt3+sqrt7) = (-9/4) * sqrt3 + (9/4) * sqrt7.

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Write the expression in simplest form. $\newline$ $(\frac{9}{\sqrt{3}+\sqrt{7}})=$

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Algebra 1

Power rule with rational exponents

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Q. Write the expression in simplest form.

\newline

(\frac{9}{\sqrt{3}+\sqrt{7}})=

Rationalize the Denominator: To simplify the expression $(\frac{9}{\sqrt{3}+\sqrt{7}})$ , we need to rationalize the denominator. This means we need to eliminate the square roots in the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of $(\sqrt{3}+\sqrt{7})$ is $(\sqrt{3}-\sqrt{7})$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator: $\frac{9}{\sqrt{3}+\sqrt{7}} \cdot \frac{\sqrt{3}-\sqrt{7}}{\sqrt{3}-\sqrt{7}}$ .
Apply Distributive Property: Use the distributive property $(a+b)(a-b) = a^2 - b^2$ to multiply the denominators: $\newline$ $(\sqrt{3}+\sqrt{7})(\sqrt{3}-\sqrt{7}) = (\sqrt{3})^2 - (\sqrt{7})^2$ .
Calculate Squares: Calculate the squares of the square roots: $(\sqrt{3})^2 - (\sqrt{7})^2 = 3 - 7$ .
Subtract Numbers: Subtract the numbers to find the value of the denominator: $3 - 7 = -4$ .
Multiply Numerator: Now, multiply the numerator by the conjugate of the denominator: $9 \times (\sqrt{3}-\sqrt{7})$ .
Divide by Denominator: Since the denominator is a constant $-4$ , we can divide the terms in the numerator by the denominator: $\frac{9 \cdot \sqrt{3}}{-4} - \frac{9 \cdot \sqrt{7}}{-4}$ .
Simplify Fractions: Simplify the fractions by dividing $9$ by $-4$ : $\left(-\frac{9}{4}\right) \cdot \sqrt{3} - \left(-\frac{9}{4}\right) \cdot \sqrt{7}.$
Write Final Expression: Write the final simplified expression: $\newline$ $(9)/(\sqrt{3}+\sqrt{7}) = (-\frac{9}{4}) \cdot \sqrt{3} + (\frac{9}{4}) \cdot \sqrt{7}.$