Rationalize the denominator. √2/√3

검증된 단계별 안내

Identify the expression that needs rationalizing: \(\frac{\sqrt{2}}{\sqrt{3}}\).

Recall that rationalizing the denominator means eliminating the square root from the denominator by multiplying numerator and denominator by the same radical.

Multiply both numerator and denominator by \(\sqrt{3}\) to get: \(\frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}\).

Use the property of radicals that \(\sqrt{a} \times \sqrt{a} = a\) to simplify the denominator: \(\sqrt{3} \times \sqrt{3} = 3\).

Multiply the numerators: \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\), so the expression becomes \(\frac{\sqrt{6}}{3}\), which has a rationalized denominator.

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주요 개념

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Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (such as square roots) from the denominator of a fraction. This is done to simplify the expression and make it easier to work with or interpret. Typically, this is achieved by multiplying the numerator and denominator by a suitable radical that will remove the root from the denominator.

Properties of Square Roots

Square roots have properties that allow simplification, such as √a × √b = √(a×b). Understanding these properties helps in manipulating expressions with radicals, especially when multiplying or dividing them. This is essential when rationalizing denominators to combine or simplify terms.

Multiplying by a Conjugate or Equivalent Radical

To rationalize denominators containing square roots, you multiply numerator and denominator by the same radical to create a perfect square in the denominator. For example, multiplying by √3/√3 removes the root in the denominator √3, since (√3)(√3) = 3, a rational number. This technique preserves the value of the expression while simplifying it.