Problem 23 - Algebra 2 and Trigonometry

Question

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \frac{\sqrt{10}+\sqrt{15}}{\sqrt{10}} $$

Step-by-Step Solution

Answer

The quotient simplifies to \( 1 + \sqrt{\frac{3}{2}} \).

Step 1: Simplify the Expression

Start with the given expression \( \frac{\sqrt{10} + \sqrt{15}}{\sqrt{10}} \). Divide each term in the numerator by \( \sqrt{10} \). The expression becomes: \( \frac{\sqrt{10}}{\sqrt{10}} + \frac{\sqrt{15}}{\sqrt{10}} \).

Step 2: Simplify Each Term Separately

Simplify \( \frac{\sqrt{10}}{\sqrt{10}} \) and \( \frac{\sqrt{15}}{\sqrt{10}} \). For the first term, \( \frac{\sqrt{10}}{\sqrt{10}} = 1 \). For the second term, rewrite \( \frac{\sqrt{15}}{\sqrt{10}} \) as \( \frac{\sqrt{15}}{\sqrt{10}} = \sqrt{\frac{15}{10}} \).

Step 3: Simplify the Square Root

Simplify \( \sqrt{\frac{15}{10}} \). This can be rewritten as \( \sqrt{\frac{3 \times 5}{2 \times 5}} = \sqrt{\frac{3}{2}} \). Therefore, the expression simplifies to \( 1 + \sqrt{\frac{3}{2}} \).

Key Concepts

Understanding Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. This can include numbers, variables, or a combination like in radical expressions. In this exercise, the fraction \( \frac{\sqrt{10} + \sqrt{15}}{\sqrt{10}} \) is an algebraic fraction because both the numerator and the denominator involve square roots, i.e., radicals.

• When simplifying algebraic fractions, we first aim to break them into simpler parts.
• This often involves dividing each term in the numerator by the denominator.
• Ensuring any common factors are canceled out is also a crucial step in simplification.

With algebraic fractions, understanding each component and how it interacts with others is essential. This exercise required dividing and simplifying, ensuring the result is in its simplest form.

Rationalizing the Denominator

Rationalizing the denominator involves removing the radical from the denominator of a fraction. This is a common practice to simplify expressions and make them easier to work with. In our problem, the denominator \( \sqrt{10} \) was originally a radical. Let's explore why rationalizing is helpful and how it's done.

• One method of rationalizing is to multiply both the numerator and the denominator by the same radical, effectively removing the radical in the denominator.
• By doing so, you convert the denominator into a rational number because multiplying radicals typically results in a whole number.

In our step-by-step solution, instead of directly rationalizing, we divided each part of the numerator by the denominator. This is an alternate route to simplify effectively, demonstrating that rationalizing isn't always necessary if simplification can be achieved through division first.

Exploring Radical Expressions

Radical expressions include expressions that contain a radical sign (√). A significant part of simplifying such expressions involves becoming comfortable with operations involving radicals, such as multiplication, division, and simplification.

• A radical expression can often be simplified by factoring out perfect squares from under the radical.
• For mixed radicals, it's essential to simplify each part separately to bring expressions to the simplest form.
• Understanding properties of radicals, like \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \), helps simplify complex expressions.

In the given exercise, after breaking down the fraction into simpler parts, we focused on simplifying \( \sqrt{\frac{15}{10}} \) further into \( \sqrt{\frac{3}{2}} \). Recognizing that a radical can often be simplified with careful factorization is key to mastering radical expressions.