Problem 44 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \left(\frac{-x}{2}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(\frac{-x}{2}\right)^{3}\) is \(\frac{-x^{3}}{8}\)

Step 1: Identify exponent

From the expression, it can be seen that the power or exponent is \(3\). Recall that when an exponent is applied to a fraction, it applies to both the numerator and the denominator. So, we need to apply the cube to \(-x\) and \(2\) separately.

Step 2: Apply exponent

Applying the exponent, \(-x\) cubed becomes \(-x * -x * -x\) which equals to \(-x^{3}\). Similarly, \(2\) cubed becomes \(2 * 2 * 2\) which equals \(8\).

Step 3: Write down the simplified expression

We can now write the expression \(\left(\frac{-x}{2}\right)^{3}\) in simplified form as \(\frac{-x^{3}}{8}\).

Key Concepts

Simplifying expressions

Simplifying expressions is a fundamental skill in algebra that involves making expressions more manageable and compact. In the exercise provided, we have the expression \( \left( \frac{-x}{2} \right)^{3} \). Our goal is to express this in the simplest form.

One of the first steps in simplifying is to look for parts of the expression that can be combined or reduced. Here, we use the rules of exponents to break down the problem.

For the expression \( \left( \frac{-x}{2} \right)^{3} \), the key point is understanding that the exponent affects both the numerator and the denominator of the fraction equally. Thus, rather than trying to multiply the whole fraction out in one step, we handle the numerator and the denominator separately. This approach simplifies the process by reducing the potential size and complexity of the numbers involved.

Remember, simplifying isn’t always about making the numbers smaller. It’s about making the math easier to understand and work with. So, keep these principles in mind while working on algebraic expressions.

Power of a fraction

When dealing with exponents in fractions, one important rule to remember is that each part of the fraction is raised to the exponent. For our example, \( \left( \frac{-x}{2} \right)^{3} \), this rule simplifies the process significantly.

It is crucial to apply the exponent to the numerator and the denominator separately:

• The numerator, \(-x\), when cubed becomes \(-x * -x * -x\). This results in \(-x^{3}\).
• The denominator, \(2\), when cubed becomes \(2 * 2 * 2\). This results in \(8\).

By handling these components individually, you convert a potentially complicated operation into smaller, easier steps, which are straightforward to compute.

This method shows how powerful and handy knowing exponent rules can be, especially when simplifying complex algebraic expressions.

Algebraic fractions

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. In the case of our problem, \( \left( \frac{-x}{2} \right)^{3} \), the fraction involves the variable \(x\) in the numerator.

Simplifying algebraic fractions requires a clear understanding of each part of the expression. Applying exponents to algebraic fractions involves using the power of a fraction principle, where each part of the fraction is raised separately to the power.

In our exercise, the expression simplifies from \( \left( \frac{-x}{2} \right)^{3} \) to \( \frac{-x^{3}}{8} \):

• The expression \(-x\) becomes \(-x^{3}\) when cubed (numerator).
• The expression \(2\) becomes \(8\) when cubed (denominator).
• Thus, the algebraic fraction simplifies to an easily readable form.

Understanding how to manipulate algebraic fractions by applying basic rules and operations is a vital skill for algebra students, as it lays the foundation for more advanced problem solving in mathematics.