Question

Simplify each expression. \(\frac{\sqrt{30}}{\sqrt{5}}\)

Short answer

\(\sqrt{6}\)

Step-by-step solution

Understand the Properties of Radicals

Recall that when you have a division of square roots, such as \(\frac{\sqrt{a}}{\sqrt{b}}\), you can write it as \(\sqrt{\frac{a}{b}}\). This property will help us simplify the expression.

Apply the Property of Radicals

Use the property from Step 1 to rewrite the expression. So we have: \(\frac{\sqrt{30}}{\sqrt{5}} = \sqrt{\frac{30}{5}}\).

Simplify the Fraction

Simplify the fraction inside the square root. Calculate \(\frac{30}{5}\) which equals 6, so we have \(\sqrt{6}\).

Write the Simplified Expression

The simplified form of the original expression \(\frac{\sqrt{30}}{\sqrt{5}}\) is \(\sqrt{6}\).

Key concepts

Properties of Radicals

When working with radicals, it's crucial to understand their fundamental properties. One key property is that the square root of a quotient can be represented as the quotient of the square roots. In simpler terms, this means that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). This property is exceptionally useful in simplifying expressions involving radicals.
This operation is valid because both square roots and division are operations that can be conducted in stages. This rearrangement can help make complicated expressions more manageable. Moreover, it gives us powerful tools to manipulate and simplify expressions involving square roots. Remember, this property only works when both \( a \) and \( b \) are positive.

Division of Square Roots

Dividing square roots involves using the property of radicals mentioned above. Whenever you have \( \frac{\sqrt{30}}{\sqrt{5}} \), you can apply the property that allows division inside the radical. Transforming into \( \sqrt{\frac{30}{5}} \) is the first step in simplification.

• This approach can often transform complex rational square root expressions into simpler radical expressions.
• It's important to ensure that the value under the square root remains valid, meaning it should be non-negative.
• This method is useful for keeping calculations neat and reducing potential errors.

Breaking down this operation makes it much easier to manage and understand the behavior of radicals during division.

Simplifying Fractions

The principle of simplifying fractions is straightforward: reduce the fraction to its simplest form. Once you've rewritten \( \sqrt{\frac{30}{5}} \), the next step is simplifying the fraction inside the radical. Here, \( \frac{30}{5} = 6 \).
Simplifying the fraction is critical because:

• It can reveal further simplification possibilities, like removing perfect squares.
• It reduces complexity and enhances calculation accuracy.

By dividing 30 by 5, you simplify the expression inside the radical to its simplest form, \( \sqrt{6} \). This process underscores the value of simplification in mathematical operations.