Rationalize the denominator. √2/√3

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.