Problem 22 - 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. $$ \frac{\sqrt{48}+\sqrt{3}}{\sqrt{3}} $$

Step-by-Step Solution

Answer

The simplified form is 5.

Step 1: Break Down the Expression

The given expression is \( \frac{\sqrt{48} + \sqrt{3}}{\sqrt{3}} \). This expression is a fraction with the numerator containing the sum of two square roots and the denominator being a single square root. Our goal is to simplify this expression as much as possible.

Step 2: Simplify Each Square Root

Simplify the square root \( \sqrt{48} \). We know that \( 48 = 16 \times 3 \), and since \( 16 \) is a perfect square, we can write \( \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \). Therefore, the expression becomes \( \frac{4\sqrt{3} + \sqrt{3}}{\sqrt{3}} \).

Step 3: Simplify the Fraction

Now that we have \( \frac{4\sqrt{3} + \sqrt{3}}{\sqrt{3}} \), notice that both terms in the numerator have \( \sqrt{3} \) in common. Factor \( \sqrt{3} \) out of the numerator: \( \sqrt{3}(4 + 1) = \sqrt{3} \times 5 \). The expression now becomes \( \frac{5\sqrt{3}}{\sqrt{3}} \).

Step 4: Cancel the Common Terms

Cancel \( \sqrt{3} \) in the numerator and the denominator. This leaves us with \( 5 \).

Step 5: Final Step: Write the Simplified Expression

The expression \( \frac{\sqrt{48} + \sqrt{3}}{\sqrt{3}} \) simplifies to \( 5 \).

Key Concepts

Understanding the Radicand

In any radical expression, such as a square root, the number or expression inside the radical sign is called the radicand. Consider the expression \( \sqrt{48} \). Here, \( 48 \) is the radicand. The radicand may also involve variables, especially in algebraic expressions.
When simplifying a radical expression, the first step often involves finding factors of the radicand that are perfect squares. This is crucial as it allows us to simplify the expression by "extracting" values from under the radical.

In the problem statement, we have \( \frac{\sqrt{48} + \sqrt{3}}{\sqrt{3}} \). Breaking down the radicand 48, it can be expressed as \( 16 \times 3 \), allowing us to simplify further since 16 is a perfect square.
By understanding and manipulating the radicand, we can simplify complex radical expressions into more manageable forms. Recognizing the factors that form the radicand is a key skill in simplifying radicals.

Spotting the Perfect Square

A perfect square is a number that can be expressed as the product of an integer with itself. In mathematical terms, a number \( n \) is a perfect square if there exists an integer \( k \) such that \( k^2 = n \).
For instance, \( 16 \) is a perfect square because it can be written as \( 4 \times 4 \) or \( 4^2 \). Recognizing these numbers is helpful when simplifying radicands.

In the simplification process of \( \sqrt{48} \), identifying the perfect square 16 is essential. We can separate \( \sqrt{48} \) into \( \sqrt{16 \times 3} \), simplifying further to \( 4\sqrt{3} \) since \( \sqrt{16} = 4 \).

• Perfect squares include numbers like 4, 9, 16, 25, and so forth.
• They become useful for extracting values from under a square root.

By effectively utilizing perfect squares, we transform seemingly complicated expressions into simpler, interpretable results.

Identifying the Common Factor

When you have a common factor in the numerator of a fraction, it often allows for further simplification of expressions. A common factor is a term that appears in both components of an algebraic expression, allowing it to be factored out.

For instance, consider the numerator \( 4\sqrt{3} + \sqrt{3} \) from our expression. Both terms include the common factor \( \sqrt{3} \). By factoring out \( \sqrt{3} \), we simplify the expression to \( \sqrt{3}(4 + 1) \) or \( 5\sqrt{3} \).
After factoring, we can then cancel this common term with a matching term in the denominator, \( \sqrt{3} \), simplifying the entire expression significantly.

Understanding how common factors work and where they appear in expressions is critical for simplifying complex mathematical problems.

• Identify the common factor in terms where possible.
• Use this factor to simplify by cancellation.

Through this method, mathematical expressions can often be reduced to their simplest forms, making them much easier to manage and solve.