Problem 3 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(7 \sqrt{12}+10 \sqrt{48}\)

Step-by-Step Solution

Answer

The simplified expression is \( 54\sqrt{3} \).

Step 1: Factor Inside the Square Roots

The first step is to factor the numbers inside the square roots to find perfect square factors. \( \sqrt{12} = \sqrt{4 \times 3} \) and \( \sqrt{48} = \sqrt{16 \times 3} \).

Step 2: Simplify the Square Roots

Extract the square roots of the perfect square factors. \( \sqrt{12} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \), and \( \sqrt{48} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \).

Step 3: Use Distribution Property

Use distribution to simplify the expression: \( 7\sqrt{12} = 7 \times 2\sqrt{3} = 14\sqrt{3} \) and \( 10\sqrt{48} = 10 \times 4\sqrt{3} = 40\sqrt{3} \).

Step 4: Combine Like Terms

Now combine like terms, \( 14\sqrt{3} + 40\sqrt{3} = (14 + 40)\sqrt{3} = 54\sqrt{3} \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This can mean many things: factorizing, cancelling terms, or combining like terms. But at its core, simplifying an expression is about making it more manageable and easier to understand.

Depending on the type of expression, the method for simplifying can vary. For expressions involving square roots, it often includes breaking down numbers into their prime factors so that we can extract square roots more easily. For example, in this exercise, the simplifying process starts by recognizing square numbers inside the expression, which are perfect squares that allow further extraction.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 times 4 equals 16. Square roots are often found in more complex expressions and need to be simplified for easier computation.

Simplifying square roots frequently involves factoring the number beneath the root into its prime components. This allows you to identify and separate perfect squares from those parts which need to remain under the square root. Take, for instance, \( \sqrt{12} \). It can be broken down into \( \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \). Recognizing and removing these perfect square factors simplifies the expression greatly.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. In this exercise, the expression \( 7\sqrt{12} + 10\sqrt{48} \) combines numbers with square roots, which are part of many algebraic expressions.

Understanding the components of algebraic expressions, including how they can be manipulated and simplified, is essential. Often, this means applying operations such as distribution or factoring to streamline the expression. This makes it more practical for use in equations or further algebraic work. Mastering the manipulation of expressions, especially involving square roots, enhances your problem-solving toolkit in mathematics.

Factoring

Factoring is the process of breaking down a number or expression into its constituent components, which when multiplied together give the original number or expression. This step is crucial when dealing with square roots, as it allows us to simplify them by isolating perfect squares.

In our example, factoring \( \sqrt{12} \) as \( \sqrt{4 \times 3} \) helps to pinpoint \( \sqrt{4} \), which we know equals 2. Thus, this extraction process is made possible through effective factoring. It's not only used in simplifying roots but also in various algebraic tasks such as finding common denominators or solving quadratic equations. In all cases, factoring breaks down complex numbers into manageable parts for easier handling.

Like Terms

Like terms are terms in an algebraic expression that have the same variables raised to the same power. They are essential in simplifying expressions because they can be combined to reduce the complexity of the expression.

For example, in the latter steps of simplifying our original expression, after simplifying the square roots, we reach \( 14\sqrt{3} + 40\sqrt{3} \). Since both terms contain \( \sqrt{3} \), they are like terms and can be combined: \((14 + 40)\sqrt{3} = 54\sqrt{3}\). This combination process relies on recognizing and uniting these like terms to arrive at a more refined and simplified expression.