Problem 29 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ 5 \sqrt{32 x}+4 \sqrt{98 x} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(48 \sqrt{2x}\).

Step 1: Simplify the Square Roots

We start by simplifying each square root separately. The square root of 32 can be simplified to \(4 \sqrt{2}\) and the square root of 98 can be simplified to \(7 \sqrt{2}\). So, the expression simplifies to \(5 \cdot 4 \sqrt{2x} + 4 \cdot 7 \sqrt{2x}\).

Step 2: Perform Arithmetic Operations

We then perform the multiplication involved in the expression: \(20 \sqrt{2x} + 28 \sqrt{2x}\).

Step 3: Combine Like Terms

Since the terms are similar, we add them: \(48 \sqrt{2x}\).

Key Concepts

Rationalizing Denominators

Rationalizing a denominator involves the process of eliminating any square roots or irrational numbers from the denominator of a fraction. This is a key skill because it helps simplify expressions, making them easier to work with.
When faced with a denominator like \( \sqrt{a} \), we can rationalize it by multiplying both the numerator and the denominator by \( \sqrt{a} \). This gives us a new expression with a rational denominator.
For example, if you have \( \frac{1}{\sqrt{2}} \), multiply both top and bottom by \( \sqrt{2} \) to get \( \frac{\sqrt{2}}{2} \).

• The goal is to make the denominator a whole number or an integer.
• Remember, rationalizing is only required for the denominator when simplifying expressions involving radicals.

A rationalized expression is often more straightforward to interpret and calculate, especially in mathematical problems that require additional operations like addition or multiplication. Rationalizing helps streamline these processes.

Square Roots

Square roots are the inverse operation of squaring a number. When you see the square root symbol \( \sqrt{} \), it asks you what number, when multiplied by itself, gives the original number under the root.
This symbol often represents just the positive square root. For instance, \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \).
Simplifying square roots is particularly important when simplifying expressions like our original exercise. To simplify square roots:

• Break down the number into its prime factors.
• Pair the factors in groups of two.
• Each pair of factors can be written outside the square root as a whole number.

For example, to simplify \( \sqrt{32} \):
Prime factors of 32 are \( 2 \times 2 \times 2 \times 2 \times 2 \). Pair these as \( (2 \times 2) \times (2 \times 2) \times 2 \), which simplifies to \( 4\sqrt{2} \).
Recognizing when and how to simplify helps tackle mathematical tasks more efficiently.

Combining Like Terms

Combining like terms is a crucial concept in simplifying algebraic expressions. Like terms are terms that have identical variables raised to the same power, though they can have different coefficients.
In the context of our problem, the expression \( 20 \sqrt{2x} + 28 \sqrt{2x} \) involves like terms. Each term has a \( \sqrt{2x} \) portion, making them combinable.
To combine like terms:

• Identify the coefficients (the numbers in front of the variables or square roots).
• Add or subtract the coefficients, keeping the variable part unchanged.

Applying this to \( 20 \sqrt{2x} + 28 \sqrt{2x} \) results in a simplified expression \( 48 \sqrt{2x} \).
Combining like terms reduces the complexity of expressions, paving the way for solving equations and further algebraic manipulations with greater ease. Mastery of this concept forms a fundamental part of algebraic proficiency.