Problem 13 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(\frac{3}{5} \sqrt{40}+\frac{5}{6} \sqrt{90}\)

Step-by-Step Solution

Answer

\(\frac{111}{30} \sqrt{10}\)

Step 1: Simplify the Square Roots

Start by simplifying the square roots in the expression. For \( \sqrt{40} \), notice that \( 40 = 4 \times 10 \). So, \( \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \).For \( \sqrt{90} \), notice that \( 90 = 9 \times 10 \). So, \( \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10} \).

Step 2: Substitute Simplified Square Roots

Replace the original square root expressions in the equation with their simplified versions: \[ \frac{3}{5} \sqrt{40} + \frac{5}{6} \sqrt{90} = \frac{3}{5} \times 2\sqrt{10} + \frac{5}{6} \times 3\sqrt{10} \]This gives us: \[ \frac{6}{5} \sqrt{10} + \frac{15}{6} \sqrt{10} \]

Step 3: Find a Common Denominator

To combine the terms, find a common denominator for \(\frac{6}{5}\) and \(\frac{15}{6}\). The least common multiple of 5 and 6 is 30.Convert the fractions:\[ \frac{6}{5} = \frac{6 \times 6}{5 \times 6} = \frac{36}{30} \]\[ \frac{15}{6} = \frac{15 \times 5}{6 \times 5} = \frac{75}{30} \]

Step 4: Combine the Terms

Now that both terms have the same denominator, add them together.\[ \frac{36}{30} \sqrt{10} + \frac{75}{30} \sqrt{10} = \left( \frac{36 + 75}{30} \right) \sqrt{10} = \frac{111}{30} \sqrt{10} \]

Step 5: Simplify the Fraction

Simplify the fraction \(\frac{111}{30}\) if possible. Since 111 and 30 have no common prime factors, the fraction is already in its simplest form. Thus, the simplified expression is \(\frac{111}{30} \sqrt{10}\).

Key Concepts

Simplifying Square Roots

Square roots can be simplified by looking for perfect square factors. Let's consider \( \sqrt{40} \) and \( \sqrt{90} \) as examples. For \( \sqrt{40} \), we notice that \( 40 = 4 \times 10 \). The number 4 is a perfect square, which means \( \sqrt{4} = 2 \). Thus, \( \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \).
Similarly, for \( \sqrt{90} \), observe that \( 90 = 9 \times 10 \). Here, 9 is a perfect square with \( \sqrt{9} = 3 \). Therefore, \( \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10} \).
By simplifying square roots this way, you can handle expressions involving square roots more easily in your calculations.

Common Denominator

When working with fractions, especially to add or subtract them, finding a common denominator is crucial. This is necessary so the fractions have a shared basis for the operation. In the case of \( \frac{6}{5} \) and \( \frac{15}{6} \), you need to determine the least common multiple (LCM) of the denominators 5 and 6.
The LCM here is 30. Convert each fraction to this common denominator:

• \( \frac{6}{5} \) becomes \( \frac{36}{30} \) (since \( 6 \times 6 = 36 \) and \( 5 \times 6 = 30 \))
• \( \frac{15}{6} \) becomes \( \frac{75}{30} \) (because \( 15 \times 5 = 75 \) and \( 6 \times 5 = 30 \))

Now with a common denominator, you can easily combine the fractions.

Fractions

Fractions are a way of representing parts of a whole. They are composed of a numerator over a denominator. Simplifying or combining fractions comes down to having a keen eye for factors. In the solution example, once you achieve a common denominator, you can add the numerators directly.
For instance, combining \( \frac{36}{30} \) and \( \frac{75}{30} \) involves summing the numerators:

• Add \( 36 + 75 \) to get \( 111 \)
• The denominator remains 30, forming \( \frac{111}{30} \)

When no further simplification is possible (no common factors other than 1), the fraction is in its simplest form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Applying the distributive property can sometimes simplify these expressions, especially when combining like terms or dealing with factors. In our example, after substituting the simplified square roots back into the fractions, the expression results in terms like \( \frac{6}{5}\sqrt{10} \).
When both terms share a common factor like \( \sqrt{10} \), you can apply the distributive property, allowing you to factor out the common element.
This simplifies the expression to a more manageable format, after which you can further simplify or combine the coefficients, as seen when reaching \( \frac{111}{30} \sqrt{10} \) from the original equation setup.