Simplify. (2)/(3+sqrt2)

Rationalize Denominator: Rationalize the denominator of the expression (2)/(3+sqrt2). To remove the square root from the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (3+sqrt2) is (3-sqrt2). (2)/(3+sqrt2) * (3-sqrt2)/(3-sqrt2)Apply Distributive Property: Apply the distributive property (also known as the FOIL method) to the denominator. (3+sqrt2)(3-sqrt2) = 3^2 - (sqrt2)^2Calculate Squares: Calculate the squares in the denominator. 3^2 - (sqrt2)^2 = 9 - 2Simplify Denominator: Simplify the denominator. 9 - 2 = 7Apply Distributive Property: Apply the distributive property to the numerator. 2(3) - 2(sqrt2) = 6 - 2sqrt2Write Simplified Expression: Write the simplified expression. The simplified form of the expression is (6 - 2sqrt2)/7.

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Math Problems

Algebra 1

Multiply radical expressions

Full solution

Q. Simplify.

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\frac{2}{3+\sqrt{2}}

Rationalize Denominator: Rationalize the denominator of the expression $\frac{2}{3+\sqrt{2}}$ . To remove the square root from the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $3+\sqrt{2}$ is $3-\sqrt{2}$ . $\frac{2}{3+\sqrt{2}} \cdot \frac{3-\sqrt{2}}{3-\sqrt{2}}$
Apply Distributive Property: Apply the distributive property (also known as the FOIL method) to the denominator. $\newline$ $(3+\sqrt{2})(3-\sqrt{2}) = 3^2 - (\sqrt{2})^2$
Calculate Squares: Calculate the squares in the denominator. $3^2 - (\sqrt{2})^2 = 9 - 2$
Simplify Denominator: Simplify the denominator. $9 - 2 = 7$
Apply Distributive Property: Apply the distributive property to the numerator. $\newline$ $2(3) - 2(\sqrt{2}) = 6 - 2\sqrt{2}$
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified form of the expression is $(6 - 2\sqrt{2})/7$ .