Problem 72 - Intermediate Algebra

Question

Factor the given factor from the expression. $$ x^{-3 / 4} ; x^{-3 / 4}+3 x^{1 / 4} $$

Step-by-Step Solution

Answer

The expression is factored as \(x^{-3/4}(1 + 3x)\).

Step 1: Identify the Common Factor

First, look at the expression \(x^{-3/4} + 3x^{1/4}\). The common factor between the two terms is \(x^{-3/4}\).

Step 2: Factor Out the Common Factor

Next, we factor out the common factor \(x^{-3/4}\) from each term in the expression. Remember that to factor a term, divide each term by the common factor. This gives us:\[x^{-3/4}(1 + 3x^{1/4 - (-3/4)})\]

Step 3: Simplify the Expression Inside the Parentheses

Simplify the expression inside the parentheses. Calculate the exponent of \(x\): \[3x^{1/4 - (-3/4)} = 3x^{1/4 + 3/4} = 3x^{4/4} = 3x^1 = 3x\] Thus, the expression becomes: \[x^{-3/4}(1 + 3x)\]

Step 4: Review the Factored Expression

The final factored expression is \(x^{-3/4}(1 + 3x)\). Ensure that each step maintains equality with the original expression.

Key Concepts

Common Factor

In the realm of algebra, one of the vital skills is identifying the common factor in an expression. A 'common factor' is a term that appears in every part of an expression or equation. It allows us to simplify problems by grouping terms together. In our exercise, the expression given is \(x^{-3/4} + 3x^{1/4}\). Here, the term \(x^{-3/4}\) is a part of both segments of the expression. By identifying \(x^{-3/4}\) as the common factor, we can then extract it from the expression, making it appear once outside of the parentheses. Performing this operation lays the groundwork for more complex simplifications by reducing the initial complexity of expressions.

Factorization

Factorization is a fundamental process in algebra where we write an expression as a product of its factors. Think of factors as building blocks, and factorizations as assembling these blocks to make up an expression. In our starting expression \(x^{-3/4} + 3x^{1/4}\), once we've identified \(x^{-3/4}\) as the common factor, the next step is performing factorization. To do this, we divide each term by the common factor (\(x^{-3/4}\)).

• Divide \(x^{-3/4}\) by itself, which gives \(1\).
• Then, divide \(3x^{1/4}\) by \(x^{-3/4}\). This requires applying exponent rules, simplifying to \(3x^{4/4}\) or \(3x\).

Hence, factoring out the common factor restructures the expression into \(x^{-3/4}(1 + 3x)\). Factorization not only simplifies calculations but also reveals important characteristics of an expression such as roots and intercepts.

Simplifying Expressions

Simplifying an expression involves making it easier to work with or understand. This often involves combining like terms, reducing the number of terms, and performing operations that streamline the expression.

After factoring an expression, like in our example of \(x^{-3/4}(1 + 3x)\), simplifying inside the parentheses is the next step. By simplifying \(1 + 3x\), we verify the exponents calculation; transforming \(3x^{4/4}\) into \(3x\). Simplifying relies heavily on the laws of exponents, such as adding the exponents when multiplying powers of the same base. Ensuring clarity and reducing the expression further can make evaluating and interpreting the algebraic expression simpler in practical applications. Simplicity doesn't just improve comprehension; it enhances accuracy by reducing the complexity of computations.