Problem 34 - Elementary Algebra Within Reach

Question

In Exercises 19-36, expand the expression as a product of factors. $$ \left(-\frac{2}{5 x}\right)^{3} $$

Step-by-Step Solution

Answer

The expression \(-(\frac{2}{5x})^3\) expands to \(-\frac{8}{125x^3}\)

Step 1: Breakdown of exponent

The main thing to know is that the exponent applies to both the numerator and the denominator of the fraction and not just one part. The formation of the expression \(-(\frac{2}{5x})^3\) is the same as \((-2)^3 / (5x)^3\).

Step 2: Calculate the powers

Calculate powers both for the numerator and the denominator independently. So, \((-2)^3\) means -2* -2 *-2 which results in -8. For the denominator, (5x)^3 means 5x * 5x * 5x which results into \(125x^3\).

Step 3: Compile the results

Bring together your results from step 2. Divide -8 by 125x^3. The final answer becomes \(-\frac{8}{125x^3}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number or an expression to a certain power. This power, or exponent, tells us how many times to multiply the base number by itself. For example, in the term \((-2)^3\), \(-2\) is the base and \(3\) is the exponent. This means we multiply \(-2\) by itself three times: \(-2 \times -2 \times -2 = -8\).

Exponentiation can apply to integers, fractions, and variables. It is a fundamental operation in algebra that allows us to express repeated multiplication compactly.
Remember:

• Base: the number or expression to be multiplied.
• Exponent: the number of times the base is multiplied by itself.

Numerator and Denominator

Understanding the components of a fraction is crucial for working with algebraic expressions. A fraction consists of two parts: the numerator and the denominator. The numerator is the top part of the fraction, indicating how many parts of the whole are being considered. The denominator is the bottom part, showing into how many equal parts the whole is divided.

In the expression \(\left(-\frac{2}{5x}\right)^3\), \(-2\) is the numerator and \(5x\) is the denominator. When we apply an exponent to a fraction, it affects both the numerator and the denominator individually. So, evaluating \(\left(-\frac{2}{5x}\right)^3\) requires us to apply the exponent separately to \(-2\) and \(5x\). This simplifies our calculations and ensures accuracy.

Power of a Fraction

When dealing with the power of a fraction, the exponent is applied to both the numerator and the denominator independently. So, if we have a fraction \(\left(\frac{a}{b}\right)^n\), it is equivalent to \(\frac{a^n}{b^n}\).

Let's look at the expression \(\left(-\frac{2}{5x}\right)^3\). We separate it into two parts:

• Numerator: \((-2)^3 = -2 \times -2 \times -2 = -8\)
• Denominator: \((5x)^3 = 5x \times 5x \times 5x = 125x^3\)

Thus, the fraction raised to a power becomes \(-\frac{8}{125x^3}\). This step helps in simplifying complex expressions and solving problems effectively.

Expansion of Expressions

Expanding expressions involves breaking them down into a series of factors, particularly when a power is involved. In the given problem \(\left(-\frac{2}{5x}\right)^3\), we expand by applying the power to each part: the numerator and the denominator.

This process:

• Helps in visualizing and simplifying complex algebraic expressions.
• Makes calculations more manageable.
• Ensures that all parts of the expression are correctly evaluated.

By expanding the expression as a product of factors, we find that \((-2)^3\) gives us \(-8\) and \((5x)^3\) yields \(125x^3\), leading to the simplified result \(-\frac{8}{125x^3}\). Understanding this process is key to mastering algebra.