Evaluate each expression or indicate that the root is not a real number.
$\frac{\sqrt7}{\sqrt3}$

Verified step by step guidance

Start with the given expression: $\frac{\sqrt7}{\sqrt3}$.

Recall that to simplify a fraction with square roots in the denominator, we rationalize the denominator by multiplying both numerator and denominator by $\sqrt3$.

Multiply numerator and denominator: $\frac{\sqrt7}{\sqrt3} \times \frac{\sqrt3}{\sqrt3} = \frac{\sqrt7 \times \sqrt3}{\sqrt3 \times \sqrt3}$.

Use the property of square roots: $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$, so the numerator becomes $\sqrt{7 \times 3} = \sqrt{21}$ and the denominator becomes $\sqrt{3 \times 3} = \sqrt{9}$.

Since $\sqrt{9} = 3$, rewrite the expression as $\frac{\sqrt{21}}{3}$, which is the simplified form.

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Key Concepts

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

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares. For example, √12 can be simplified to 2√3 because 12 = 4 × 3 and √4 = 2. This process helps in making expressions easier to work with and compare.

Rationalizing the Denominator

Rationalizing the denominator means eliminating any radicals from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable radical to create a rational number in the denominator. For example, to rationalize 1/√3, multiply by √3/√3 to get √3/3.

Properties of Square Roots and Fractions

The property √a/√b = √(a/b) allows combining or separating radicals in fractions. Understanding this helps in rewriting expressions like (√7)/(√3) as √(7/3), which can then be simplified or rationalized. This property is fundamental in manipulating radical expressions.

Evaluate each expression or indicate that the root is not a real number.73\(\frac{\sqrt7}{\sqrt3}\)37