Problem 71 - Intermediate Algebra

Question

Factor the given factor from the expression. $$ x^{-1 / 3} ; 5 x^{-1 / 3}+x^{2 / 3} $$

Step-by-Step Solution

Answer

The expression factors to \( x^{-1/3}(5 + x) \).

Step 1: Identify the Common Factor

Examine the expression \( 5x^{-1/3} + x^{2/3} \) and identify the common factor. In this case, the common factor is \( x^{-1/3} \).

Step 2: Factor out the Common Factor

Factor \( x^{-1/3} \) out of each term in the expression. This is done by dividing each term by \( x^{-1/3} \): 1. \( 5x^{-1/3} \) divided by \( x^{-1/3} \) is \( 5 \).2. \( x^{2/3} \) divided by \( x^{-1/3} \) is \( x^{(2/3) - (-1/3)} = x^{3/3} = x^1 = x \). Thus, factoring out \( x^{-1/3} \), we get \( x^{-1/3}(5 + x) \).

Step 3: Combine the Expression

Write the factored expression in its final form: \( x^{-1/3}(5 + x) \).

Key Concepts

Common Factor

A common factor in algebraic expressions is a term that appears in each part of the expression. Imagine if two terms, like apples and oranges, have something in common, such as being fruits. When working with algebraic terms, we look for a mathematical component they share. In the expression \(5x^{-1/3} + x^{2/3}\), the term \(x^{-1/3}\) is a component that both parts of the expression share.

This process is called "factoring out." It involves pulling out this common term so that you can rewrite the expression in a simplified way. Spotting the common factor is the first step to simplifying complex algebraic expressions, making them easier to manage and work with.

Algebraic Expressions

An algebraic expression is much like a sentence in math. Instead of using words, we use numbers, variables, and operators like \(+\) or \(-\). For example, \(5x^{-1/3} + x^{2/3}\) is an algebraic expression. Each part of this expression (\(5x^{-1/3}\) and \(x^{2/3}\)) is called a term.

To effectively work with algebraic expressions, it's essential to understand each component:

• **Terms**: Parts of the expression that are separated by \(+\) or \(-\).
• **Variables**: Symbols like \(x\) that represent unknown values.
• **Coefficients**: Numbers like \(5\) that multiply the variables.

Combining these elements allows us to build expressions and solve equations systematically.

Negative Exponents

Negative exponents might seem tricky at first, but they are simply another way to express division in mathematical terms. A negative exponent tells you to take the reciprocal, or "flip," of the base number.

For instance, \(x^{-1/3}\) means \(1/x^{1/3}\). It indicates you need to divide 1 by \(x^{1/3}\). Remember:

• \(a^{-n} = 1/a^n\)
• \(x^{-1/3} = 1/x^{1/3}\)

This means when you see a negative exponent, you aren't dealing with negative numbers, just fractions, which can make expressions smaller or simpler.

Factoring Out

Factoring out involves finding a common factor in all terms of an expression, then rewriting the expression in a simpler form. For example, when you "factor out" the common factor \(x^{-1/3}\) from \(5x^{-1/3} + x^{2/3}\), you are essentially dividing each term by \(x^{-1/3}\).

Here's how it works:

• Divide \(5x^{-1/3}\) by \(x^{-1/3}\), which simplifies to \(5\).
• Divide \(x^{2/3}\) by \(x^{-1/3}\), which leaves \(x^{(2/3) - (-1/3)} = x^{3/3} = x\).

By factoring out \(x^{-1/3}\), we end up with the expression \(x^{-1/3}(5 + x)\). This operation makes it easier to handle and work with algebraic expressions in calculations.