Problem 67 - Intermediate Algebra

Question

Factor the given factor from the expression. $$ x^{8 / 3} ; x^{8 / 3}+x^{10 / 3} $$

Step-by-Step Solution

Answer

The expression factors to \(x^{8/3}(1 + x^{2/3})\).

Step 1: Identify the Common Factor

Observe that both terms in the expression \(x^{8/3} + x^{10/3}\) share the factor \(x^{8/3}\). Identify this as the common factor that needs to be factored out from the expression.

Step 2: Factor Out the Common Factor

Factor out \(x^{8/3}\) from the expression. This means re-writing the expression as \(x^{8/3}(1 + x^{10/3 - 8/3})\).

Step 3: Simplify the Exponents

Calculate \(x^{10/3 - 8/3}\). By subtracting the exponents, we get \(x^{2/3}\). So, the expression becomes \(x^{8/3}(1 + x^{2/3})\).

Step 4: Write the Final Factored Expression

The expression is now factored, and the final result is \(x^{8/3}(1 + x^{2/3})\).

Key Concepts

Common Factor

When factoring algebraic expressions, identifying the common factor is often the first crucial step. A common factor is a term that divides all components of a given expression evenly. In the expression \(x^{8/3} + x^{10/3}\), both terms share \(x^{8/3}\) as a common factor. This means that \(x^{8/3}\) can be factored out of the entire expression.

Finding a common factor is like looking for a common thread that binds all terms. In algebra, this typically involves:

• Identifying any coefficients that multiply all terms.
• Recognizing any common variable bases in terms with exponents, and taking the lowest power shared.

In the given exercise, the commonality arises from having the same base \(x\) with fractional exponents. Factoring this out simplifies the expression, making it easier to handle and often necessary for further algebraic operations.

Exponent Rules

Understanding and applying exponent rules is fundamental when dealing with algebraic expressions that involve powers. One of these rules is the subtraction of exponents when dividing like bases. For example, when you have a term like \(x^{10/3}\) and you factor out \(x^{8/3}\), you need to subtract these exponents according to the rule:

• \(x^a / x^b = x^{a-b}\)

When we apply this rule to \(x^{10/3}/x^{8/3}\), we perform the subtraction of exponents: \[x^{10/3 - 8/3} = x^{2/3}\]This operation leaves us with \(x^{2/3}\), simplifying the initial expression to:\[x^{8/3} (1 + x^{2/3})\]

Exponent rules help:

• Simplify expressions by reducing the complexity of exponents.
• Ensure consistency in calculations involving powers.

Mastering these rules allows for efficient manipulation of algebraic expressions.

Simplifying Expressions

Once you've factored out the common factor from an expression, the next step is to simplify what's left. Simplifying involves reducing the expression to its simplest form while maintaining equivalent relationships. In our example, after factoring out \(x^{8/3}\), we were left with:
\(x^{8/3} (1 + x^{2/3})\)

This means:

• Checking if further simplification is possible in the polynomial inside the parentheses.
• Ensuring all components inside any brackets are in their simplest terms.

In this case, the expression \(1 + x^{2/3}\) cannot be simplified further. Thus, the factored expression is considered fully simplified.

Simplifying expressions is the goal to achieve a form that’s easier to understand or work with in further mathematical operations. This often involves:

• Combining like terms.
• Applying algebraic identities.

Simplifying not only aids during computation but also helps in verifying results and finding errors in more complex expressions.