Problem 55 - College Algebra

Question

Simplify each exponential expression $$ \left(4 x^{3}\right)^{-2} $$

Step-by-Step Solution

Answer

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

Step 1: Rules of negative exponents

First, it's useful to understand the rules that govern negative exponents. Generally, if \( a \) is any non-zero real number and \( n \) is any integer, then \( a^{-n} = \frac{1}{a^{n}} \). So, \( (4x^3)^{-2} \) becomes \( \frac{1}{{(4x^3)}^{2}} \)

Step 2: Apply the rules of exponents

Applying the rule of exponents \( \left(a^m\right)^n = a^{mn} \), we can simplify \( \frac{1}{{(4x^3)}^{2}} \) to \( \frac{1}{4^{2} \cdot (x^{3})^{2}} \). This further simplifies to \( \frac{1}{16x^{6}} \)

Key Concepts

Negative Exponents

When dealing with negative exponents, it's important to understand what they signify. A negative exponent indicates that the base should be moved from one part of a fraction to the other. More simply, a negative exponent represents the reciprocal of the base raised to the positive version of that exponent. For example:

• If you have an exponential expression like \( a^{-n} \), it is equivalent to \( \frac{1}{a^n} \).
• This means that \( a^{-n} \) "flips" the base \( a \) to the denominator of a fraction and makes the exponent positive.

In the given problem \( (4x^3)^{-2} \), the negative exponent \(-2\) tells us to take the reciprocal of \( (4x^3)^2 \). Hence, we convert it to \( \frac{1}{(4x^3)^2} \), marking the first step of simplification.

Rules of Exponents

The rules of exponents are vital for efficiently simplifying expressions involving powers. Here are some key rules:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)

In our example, to simplify \( \frac{1}{(4x^3)^2} \), we apply the "power of a power" rule. This involves multiplying the exponents, turning the expression into \( \frac{1}{4^2 \times (x^3)^2} \). Through these rules, each part of the expression can be simplified separately. Understanding these rules allows for a quicker and more straightforward simplification process.

Simplifying Expressions

Simplifying expressions is the process of reducing complexity while keeping the expression equivalent. In algebra, this often involves combining like terms, applying rules, or simplifying fractions and powers.

• Initially, we had \( (4x^3)^{-2} \), which we rewritten using the negative exponent rule into \( \frac{1}{(4x^3)^2} \).
• Next, using the rules of exponents, we broke it down to \( \frac{1}{4^2 \cdot (x^3)^2} \).
• By calculating the powers individually, it becomes \( \frac{1}{16x^6} \).

Each of these steps simplifies the expression down to its most basic form. The key is consistently applying each step while following the arithmetic and algebraic rules. Simplifying expressions is a fundamental skill that makes complex algebraic problems much more manageable.