Problem 73 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. All variables represent positive real numbers. \(-3 \sqrt{2 x^{3}}+4 \sqrt{8 x^{3}}-3 \sqrt{32 x^{3}}\)

Step-by-Step Solution

Answer

The expression simplifies to \(-7 \sqrt{2} x^{3/2}\).

Step 1: Simplify Each Radical Expression

First, simplify the square root of each term. For the first term, \(\sqrt{2x^3} = \sqrt{2} \cdot x^{3/2}\). For the second term, \(\sqrt{8x^3} = \sqrt{4 \cdot 2} \cdot \sqrt{x^3} = 2 \sqrt{2} \cdot x^{3/2}\). For the third term, \(\sqrt{32x^3} = \sqrt{16 \cdot 2} \cdot \sqrt{x^3} = 4 \sqrt{2} \cdot x^{3/2}\).

Step 2: Apply the Distributive Property

Use the distributive property to express each term in the form \(a \sqrt{2} x^{3/2}\) where \(a\) is a coefficient. This gives us \(-3 \sqrt{2} x^{3/2} + 4 \cdot 2 \sqrt{2} x^{3/2} - 3 \cdot 4 \sqrt{2} x^{3/2}\).

Step 3: Simplify Coefficients

Calculate the coefficients for each term: \(-3 \sqrt{2} x^{3/2}, \ 8 \sqrt{2} x^{3/2} \) for the second term, and \(-12 \sqrt{2} x^{3/2}\) for the third term.

Step 4: Combine Like Terms

Combine all the terms: \(-3 \sqrt{2} x^{3/2} + 8 \sqrt{2} x^{3/2} - 12 \sqrt{2} x^{3/2} = (-3 + 8 - 12) \sqrt{2} x^{3/2} = -7 \sqrt{2} x^{3/2}\).

Key Concepts

Radical Expressions

When dealing with radical expressions, it's crucial to understand what they represent. A radical expression involves a root symbol, such as the square root or cube root. In mathematics, a square root tries to determine what number multiplied by itself gives a specific value.
For instance, the expression \( \sqrt{2x^3} \) combines a numerical square root with variables. It indeed contains two parts: the numerical part (the number) and the variable part (with the exponent).
To work with radical expressions effectively:

• Simplify them where possible. This often involves breaking down the number inside the radical into its factors.
• Look for perfect squares or cubes that can be extracted from under the radical sign.
• Understand that if two radicals share the same radicand, you can combine them through operations like addition or subtraction.

Mastering radical expressions allows you to simplify complex algebraic problems and find solutions more easily.

Simplification

Simplification in mathematics means reducing an expression to its simplest form. It's about making equations easier to handle while maintaining their value.
In the given exercise, the simplification involves reducing each term like \( \sqrt{8x^3} \) to its simplest components. This requires you to:

• Factor out perfect squares or cubes. For example, \( 8 \) can be rewritten as \( 4 \times 2 \), where \( 4 \) is a perfect square \((2^2)\).
• Simplify further using exponent rules. If you have \( x^3 \), it can be broken down as \( x^{3/2} \times x^{3/2} \).
• Recombine all simplified parts to form a reduced expression.

The goal is to express every term with the minimal radical, enabling you to apply other algebraic operations, such as addition or subtraction easily. This reduced form is vital, especially when moving to the step of combining like terms, which simplifies the entire equation further.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In the exercise, we work with expressions containing variables represented by positive real numbers.
Understanding algebraic expressions means:

• Recognizing like terms. These are terms whose variables and their exponents are identical.
• Using operations like addition and subtraction to simplify or alter expressions.
• Applying the distributive property, which allows you to multiply across terms within parentheses, aiding in further simplification.

For instance, the distributive property helped rewrite the terms with a common factor \( \sqrt{2} x^{3/2} \), making it easier to combine them. Algebraic expressions allow you to model real-world situations and solve problems by manipulating numbers and variables efficiently.