Problem 2 - Intermediate Algebra

Question

Fill in the blanks. To multiply \(2 \sqrt{5}(3 \sqrt{8}+\sqrt{3}),\) use the _____ property.

Step-by-Step Solution

Answer

Use the distributive property.

Step 1: Identify the Property

When multiplying expressions like \(2 \sqrt{5}(3 \sqrt{8}+\sqrt{3})\), we use the distributive property. This property allows us to distribute one term across the terms inside the bracket.

Step 2: Apply the Distributive Property

According to the distributive property: \( a(b + c) = ab + ac \). Therefore, multiply \(2 \sqrt{5}\) by each term inside the parentheses separately: \(2 \sqrt{5} \times 3 \sqrt{8}\) and \(2 \sqrt{5} \times \sqrt{3}\).

Step 3: Multiply Each Pair of Terms

First, multiply \(2 \sqrt{5} \times 3 \sqrt{8}\): - Coefficients: \(2 \times 3 = 6\). - Square root terms: \(\sqrt{5} \times \sqrt{8} = \sqrt{40}\).Thus, we have \(6 \sqrt{40}\).

Step 4: Simplify \(\sqrt{40}\)

Simplify \(\sqrt{40}\) by factoring it into \(\sqrt{4 \times 10}\). Since \(\sqrt{4} = 2\), we get \(2 \sqrt{10}\). Thus, \(6 \sqrt{40} = 6 \times 2 \sqrt{10} = 12 \sqrt{10}\).

Step 5: Multiply Second Pair of Terms

Multiply \(2 \sqrt{5} \times \sqrt{3}\):- Since there is no coefficient besides \(1\), just multiply the roots: \(\sqrt{5} \times \sqrt{3} = \sqrt{15}\), giving us \(2 \sqrt{15}\).

Step 6: Sum the Results

Add the two products obtained from the distribution:\(12 \sqrt{10} + 2 \sqrt{15}\).

Key Concepts

Multiplying Radical Expressions

When multiplying radical expressions, the distributive property becomes a valuable tool. This property states that multiplying a single term by terms inside parentheses is done individually. Specifically, if you have an expression like \(a(b + c)\), it results in \(ab + ac\). Using the distributive property, each term in the parentheses is multiplied separately by the term outside.

In the multiplication \(2 \sqrt{5}(3 \sqrt{8} + \sqrt{3})\), follow these steps:

• First, multiply \(2 \sqrt{5}\) by \(3 \sqrt{8}\). Do this by multiplying the coefficients (2 and 3) and then the radical parts \(\sqrt{5}\) and \(\sqrt{8}\).
• Then, multiply \(2 \sqrt{5}\) by \(\sqrt{3}\), following the same logic.

This method breaks down the problem, making it easier to handle. Thus, the distributive property not only simplifies our work but also ensures precise calculations when dealing with multiple terms.

Simplifying Radicals

Sometimes working with radicals requires simplification for easier solutions. Simplifying radicals involves breaking down a radical to its most basic form. The goal is to factor the number into a product of perfect squares and simplify.

Let's take the operation \(\sqrt{40}\) as an example:

• Start by factoring 40 into its prime factors. In this case, 40 can be factored as \(4 \times 10\).
• Identify the perfect squares. Here \(4\) is a perfect square (\(2^2\)).
• Simplify \(\sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10}\). Knowing \(\sqrt{4} = 2\), rewrite the expression as \(2 \sqrt{10}\).

Through this simplification, \(6 \sqrt{40}\) becomes \(12 \sqrt{10}\), making calculations more consistent and clearer. Simplifying helps not only in accuracy but also speeds up solving.

Square Roots

Understanding square roots is crucial when working with radicals. The square root of a number \(x\) is a number \(y\) such that \(y^2 = x\). Every positive number has two square roots: one positive and one negative, but by convention, we use the positive version in calculations.

Here's a quick breakdown:

• \(\sqrt{9} = 3\) because \(3^2 = 9\).
• Any positive number can be broken into a product of factors, helping to determine its square root.

When combining square roots, multiplication follows: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). Thus, \(\sqrt{5} \times \sqrt{3} = \sqrt{15}\). This principle helps expand our expertise in multiplying radicals and ensures a clear understanding of operations involved. Whether it's through distribution or simplification, these principles guide calculations with precision.