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Simplify radical expressions by rationalizing the denominator

# Simplify radical expressions by rationalizing the denominator

To simplify radical expressions by rationalizing the denominator it will be useful to remember the following properties.

• The multiplication property of square roots states that

$sqrt{ab}=\;sqrt{a}sqrt{b}$

• The division property of square roots states that

$sqrt{\frac{a}{b}}=\;\frac{sqrt{a}}{sqrt{b}}$

Simplify $sqrt{\frac{5}{3}}$

To simplify this radical expression, rewrite it with a radical in the numerator and a radical in the denominator. Then factor out any perfect squares and simplify.

$sqrt{\frac{5}{3}}$
=$sqrt{\frac{5}{3}}$   Find the prime factorizations of the numerator and denominator
=$\frac{sqrt{5}}{sqrt{3}}$   Apply the division property of square roots

To finish simplifying the expression, multiply by $\frac{sqrt{3}}{sqrt{3}}$ to rationalize the denominator.

$\frac{sqrt{5}}{sqrt{3}}\;\cdot\;\frac{sqrt{3}}{sqrt{3}}$

=$\frac{sqrt{5\cdot3}}{(sqrt{3})^2}$

Simplify
=$\frac{sqrt{5\cdot3}}{3}$

Simplify the denominator
=$\frac{sqrt{15}}{3}$   Multiply

Simplify.

$\sqrt{\frac{7}{5}$

$\frac{Array}{\sqrt{}}Solution\;=$