Problem 84 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{64 x^{3}}-\sqrt{x}+3 \sqrt{x} $$

Step-by-Step Solution

Answer

The simplified expression is \( 8x^{3/2} + 2\sqrt{x} \).

Step 1: Simplify Under Square Roots

First, simplify the square root expressions. Start with \( \sqrt{64x^3} \). The square root of a product can be separated into the product of square roots: \( \sqrt{64} \cdot \sqrt{x^3} \). Since \( \sqrt{64} = 8 \) and \( \sqrt{x^3} = x^{3/2} \), this gives us \( 8x^{3/2} \).

Step 2: Simplify Expression

Now substitute the simplified expression from Step 1 into the original expression. Thus, we have:\( 8x^{3/2} - \sqrt{x} + 3\sqrt{x} \).

Step 3: Combine Like Terms

The terms \( -\sqrt{x} \) and \( 3\sqrt{x} \) are like terms. They can be combined by adding their coefficients:\( -1\sqrt{x} + 3\sqrt{x} = 2\sqrt{x} \).Thus, the expression becomes:\( 8x^{3/2} + 2\sqrt{x} \).

Key Concepts

Square Roots

Understanding square roots is essential for simplifying many algebraic expressions. The square root function essentially undoes the squaring process. In simpler terms, the square root of a number is a value that, when multiplied by itself, gives the original number.

In expressions, the square root can be distributed over multiplication. So, for a product inside a square root, like \( \sqrt{ab} \), this can be split into \( \sqrt{a} \cdot \sqrt{b} \).

• This property allows for simplification where each part under the square root is considered separately.
• For instance, take \( \sqrt{64x^3} \): It's solved by \( \sqrt{64} \) and \( \sqrt{x^3} \).

Knowing that \( \sqrt{64} = 8 \), because \( 8 \times 8 = 64 \),

• and using exponent rules to handle \( \sqrt{x^3} \), which becomes \( x^{3/2} \), can greatly simplify your task.

By breaking the square root operations into known square roots, simplifying expressions involving square roots becomes much more manageable.

Combining Like Terms

Combining like terms is a key skill in simplifying algebraic expressions. Like terms are terms that have the same variable part. They can be combined by adding or subtracting their coefficients.

In our exercise, observe that \( -\sqrt{x} \) and \( 3\sqrt{x} \) have the same variable part, \( \sqrt{x} \). So, they can be combined into one single term.

• Think of the coefficients: the number directly in front of \( \sqrt{x} \).
• The expression has \( -1 \cdot \sqrt{x} \) and \( 3 \cdot \sqrt{x} \).
• Adding their coefficients yields \( -1 + 3 \), resulting in \( 2\sqrt{x} \).

Combining terms reduces complexity and reveals useful structures in the expression. By simplifying terms, you create equations that are easier to work with for further mathematical operations or problem-solving.

Algebraic Simplification

Algebraic simplification involves performing operations that reduce an expression to its simplest form while maintaining its equivalence. The ability to simplify expressions is foundational for solving more complex algebraic problems.

The process typically includes:

• Simplifying each component, like square roots or exponents.
• Combining like terms to reduce the length of expressions.

Let's look at our initial expression:
\[ \sqrt{64x^3} - \sqrt{x} + 3\sqrt{x} \]. After simplifying the square roots, you have \( 8x^{3/2} - \sqrt{x} + 3\sqrt{x} \).

• Combine the current like terms \( -\sqrt{x} \) and \( 3\sqrt{x} \) to yield \( 2\sqrt{x} \).

Leaving us with a compact expression:
\[ 8x^{3/2} + 2\sqrt{x} \].
Algebraic simplification is not just about making equations look prettier. It makes calculations easier and helps reveal underlying relationships between variables within the expression.