Problem 78 - Algebra 2

Question

Simplify each expression. \(\frac{\sqrt{7}}{\sqrt{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{\sqrt{14}}{2}\).

Step 1: Apply the Quotient Rule for Radicals

The expression given is \(\frac{\sqrt{7}}{\sqrt{2}}\). According to the quotient rule for radicals, \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\). Thus, we can rewrite the expression as \(\sqrt{\frac{7}{2}}\).

Step 2: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{2}\) to eliminate the square root in the denominator. This gives \(\frac{\sqrt{7} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{14}}{2}\). The denominator \(\sqrt{2}\times \sqrt{2}\) simplifies to 2.

Step 3: Simplify the Expression

No further simplification can be done for \(\frac{\sqrt{14}}{2}\) because \(\sqrt{14}\) is in its simplest form, and the fraction is also simplified. Thus, the simplified expression is \(\frac{\sqrt{14}}{2}\).

Key Concepts

Quotient Rule for Radicals

The quotient rule for radicals is a powerful tool when simplifying expressions involving roots. It allows us to combine or separate roots in a fraction. The basic principle is that \( \frac{\sqrt{a}}{\sqrt{b}} \) is equivalent to \( \sqrt{\frac{a}{b}} \). This means you can either keep the radicals separate as a fraction or combine them under a single square root.

• This rule is particularly helpful in simplifying calculations. Instead of performing two separate square root operations, one for the numerator and one for the denominator, you can perform a single square root operation on the fraction.
• It simplifies expressions with square roots by writing them in an alternate form, which might make further steps easier.

Consider the original expression: \( \frac{\sqrt{7}}{\sqrt{2}} \). Applying the quotient rule, we rewrite it as \( \sqrt{\frac{7}{2}} \). This new form lays the groundwork for further simplification.

Simplifying Radicals

Simplifying radicals often involves breaking down the number inside the root into its simplest form and ensuring the expression is as concise as possible. When dealing with radicals, it's important to recognize factors that can be reformulated into simpler powers.

• If the number under the square root can be factored into perfect squares, it could be split further.
• Using prime factorization helps identify these perfect squares, making it easier to "take out" the square root.

In our example after rationalizing \( \frac{\sqrt{7} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} \), we ended up with \( \frac{\sqrt{14}}{2} \). Here, \( \sqrt{14} \) is already in its simplest form since 14 factors as 2 and 7, neither of which are perfect squares. Therefore, there are no further simplifications needed.

Radicals in Fractions

Dealing with radicals in fractions usually aims to eliminate the radical from the denominator through a process known as rationalization. This process makes the denominator a rational number, typically by multiplying the numerator and the denominator of the fraction by a suitable term.

• Rationalizing the denominator means that you multiply the fraction by \( \frac{\sqrt{b}}{\sqrt{b}} \) if the denominator is \( \sqrt{b} \), effectively removing the radical from the denominator.
• Ensure that you multiply both the numerator and the denominator with the same radical to maintain the value of the expression.

In our exercise: \( \frac{\sqrt{7}}{\sqrt{2}} \) was rationalized by multiplying by \( \frac{\sqrt{2}}{\sqrt{2}} \), resulting in \( \frac{\sqrt{14}}{2} \). This transformation removed the radical from the denominator—making the expression easier to handle in further mathematical operations or comparisons.