Problem 13 - Technical Mathematics with Calculus

Question

Simplify, and write without negative exponents. Do some by calculator. $$\left(4 a^{3} b^{2} c^{6}\right)^{-2}$$

Step-by-Step Solution

Answer

\(\frac{1}{16 a^{6} b^{4} c^{12}}\)

Step 1: Apply the Negative Exponent Rule

First, apply the negative exponent rule, which states that any term with a negative exponent can be rewritten as the reciprocal of the term with a positive exponent. In this case, we can rewrite the entire expression as the reciprocal of itself to the power of positive 2: \[ \(4 a^{3} b^{2} c^{6}\)^{-2} = \frac{1}{\(4 a^{3} b^{2} c^{6}\)^2} \.\]

Step 2: Apply the Power to a Power Rule

Next, apply the power to a power rule, which states that when you raise a power to another power, you multiply the exponents. Here, we multiply the exponents of each variable inside the parentheses by 2: \[ \frac{1}{\(4 a^{3} b^{2} c^{6}\)^2} = \frac{1}{4^2 a^{3\cdot2} b^{2\cdot2} c^{6\cdot2}} \]

Step 3: Simplify the Expression

Simplify the expression by calculating the exponents: \[ \frac{1}{4^2 a^{3\cdot2} b^{2\cdot2} c^{6\cdot2}} = \frac{1}{16 a^{6} b^{4} c^{12}} \.\] This is the simplified expression without negative exponents.

Key Concepts

Negative Exponent Rule

Understanding the negative exponent rule is essential for simplifying algebraic expressions that include terms raised to negative powers. In mathematics, an expression such as \( a^{-n} \) can be intimidating at first, but the rule to simplify it is straightforward: a negative exponent indicates that the term can be transformed into a reciprocal raised to the corresponding positive exponent. That is, \( a^{-n} = \frac{1}{a^n} \).

This powerful rule tells us that instead of thinking about negative exponents as 'negative' operations, we can view them as an instruction to invert the base term. For example, \( 2^{-3} \) is the same as \( \frac{1}{2^3} \), or \( \frac{1}{8} \). Applying the negative exponent rule is the first step in simplifying complex expressions and lays the groundwork for further simplification.

Power to a Power Rule

Once negative exponents are handled, expressions often require further simplification, particularly when they contain powers raised to additional powers. This is where the power to a power rule comes into play. It states that when you take a power and raise it to another power, you multiply the exponents together. This is mathematically represented as \( (a^n)^m = a^{n \times m} \).

Applying this rule is pivotal in simplifying expressions to their most basic form. For instance, consider \( (x^2)^3 \); according to the power to a power rule, this is equal to \( x^{2 \times 3} = x^6 \). It is important to remember that this rule applies to every term within the parentheses, so all bases and their exponents are subject to this multiplication step. This can dramatically change the form of an expression and is a necessary procedure before arriving at the final, simplified result.

Simplifying Algebraic Expressions

Simplification is the process of converting complex algebraic expressions into the most reduced form possible. Skilled simplification involves applying rules such as the negative exponent rule and the power to a power rule comprehensively. Alongside these, combining like terms and carrying out arithmetic operations are important strategies.

Take for instance the expression \( \frac{1}{16 a^{6} b^{4} c^{12}} \). Simplification isn't just about applying rules; it's also about recognizing that no further reduction is possible in this case, as each term is in its simplest form with no like terms to combine. A pivotal goal of simplifying algebraic expressions is to make them easier to evaluate, compare, and use in further calculations, including in-field applications like physics and engineering. Striving for simplicity can often reveal underlying patterns and relationships within mathematical concepts.