Problem 55 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(4 x^{3}\right)^{-2}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{1}{16 x^{6}} \).

Step 1: Recognize the Power Rule

The rule we can apply here is the power rule, which states that \((a^{m})^{n} = a^{mn}\). So, we need to raise \(4 x^{3}\) to the power of -2. It means first we need to take a reciprocal of \(4 x^{3}\) and then square it.

Step 2: Apply the Power Rule

The expression becomes \( \left(\frac{1}{4 x^{3}}\right)^{2}\). Now squaring a fraction implies squaring both the numerator and the denominator.

Step 3: Square the Fraction

We square the fraction to get \( \left(\frac{1}{4 x^{3}}\right)^{2} = \frac{1^{2}}{(4 x^{3})^{2}} \).

Step 4: Simplify the Expression

One squared is still 1, and for the denominator we use the same power rule as in step 1 to the expression \(4^{2} (x^{3})^{2}\) to get \( \frac{1}{16 x^{6}} \).

Key Concepts

Understanding the Power Rule

The power rule is essential when working with exponential expressions. It simplifies operations where you raise an exponential expression to another power.
In mathematical terms, the power rule is expressed as \((a^{m})^{n} = a^{m \cdot n}\). This means if you raise a power to another power, you multiply the exponents.

This rule is extremely handy for simplifying complex exponential expressions like \((4x^3)^{-2}\). Here, it's crucial to know that the exponent "-2" affects the entire expression in the parentheses.

• For \(4x^3\), the expression already has an exponent of 3 on \(x\).
• Applying the power rule involves multiplying the inner exponent by -2 to get \((4^1 \cdot x^{3}) ^{-2}\).

Recognizing and correctly utilizing the power rule helps you transform difficult expressions into more manageable forms.

What is a Reciprocal?

Reciprocals play a vital role in simplifying exponential expressions, especially those with negative exponents.

To find the reciprocal of a number or expression, you essentially "flip" it. For instance, the reciprocal of \(a\) is \(\frac{1}{a}\). This inverse relationship helps manage negative exponents.

Take \((4x^3)^{-2}\) as an example. The negative exponent signals the need for a reciprocal.

• Transform the expression into its reciprocal to rewrite it as \(\left(\frac{1}{4 x^{3}}\right)^{2}\).
• The negative exponent has now been addressed by the reciprocal step, ensuring all subsequent steps can focus on positive powers.

Reciprocals effectively simplify negative exponents by turning them positive, streamlining the expression.

Mastering Fraction Simplification

Fraction simplification is crucial when evaluating and reducing expressions involving fractions. It involves breaking down complex numerators or denominators to their simplest form.

After changing \((4x^3)^{-2}\) to \(\left(\frac{1}{4 x^{3}}\right)^{2}\), you must square the fraction:

• Square both the numerator and the denominator to get \(\frac{1^2}{(4 x^3)^2}\).
• The numerator, \(1^2\), remains 1.
• The denominator becomes \(4^2 \cdot (x^3)^2\) equals \(16x^6\).

The final simplified result is \(\frac{1}{16x^6}\).
By systematically applying fraction simplification, you accurately reduce expressions to their simplest forms.