Problem 29 - Technical Mathematics with Calculus

Question

Challenge Problems. $$(3 m)^{-3}-2 n^{-2}$$

Step-by-Step Solution

Answer

\(\frac{1}{27m^3} - \frac{2}{n^2}\)

Step 1: Simplify the expression with a negative exponent (first term)

To deal with a negative exponent, recognize that a number raised to a negative exponent is the reciprocal of that number raised to the opposite positive exponent. This means that \((3m)^{-3}\) can be rewritten as \(\frac{1}{(3m)^3}\). Expand the denominator using the power rule \((a^b)^c = a^{b*c}\), which gives us \(\frac{1}{27m^3}\).

Step 2: Simplify the expression with a negative exponent (second term)

Similarly, for the second term \(-2n^{-2}\), apply the negative exponent rule. Write it as \(-2 * \frac{1}{n^2}\) or \(-\frac{2}{n^2}\).

Step 3: Combine both terms into a single expression

Combine the two terms you've now simplified, which gives the final expression \(\frac{1}{27m^3} - \frac{2}{n^2}\). This is the simplified version of the original expression.

Key Concepts

Understanding the Power Rule in Algebra

The power rule is a fundamental principle in algebra that helps simplify expressions involving exponents. It states that when an expression with an exponent is raised to another power, you can multiply the exponents. Formally, the power rule is expressed as \( (a^b)^c = a^{b \times c} \).

For example, in the expression \( (3m)^{-3} \), applying the power rule involves multiplying the exponents of 3 and \( m \) individually by -3. This leads to \( 3^{-3} \) and \( m^{-3} \) respectively, eventually simplifying down to \( \frac{1}{27m^3} \), after converting negative exponents to positive ones. It's useful to remember that negative exponents indicate the reciprocal of the base raised to the positive exponent. The power rule is a valuable tool in simplifying complex algebraic expressions and is frequently used alongside other properties of exponents.

Simplifying Expressions with Negative Exponents

Simplifying expressions is a critical skill in algebra that involves reducing expressions to their simplest form. When encountering negative exponents, one must remember that they represent the reciprocal of the base with a positive exponent. In the context of our example \( (3m)^{-3} \) and \( -2n^{-2} \), simplification involves converting terms with negative exponents to their reciprocal forms.

The expression \( (3m)^{-3} \) simplifies to \( \frac{1}{(3m)^3} \) by flipping the base and making the exponent positive. This interpretation is crucial as it reflects the underlying concept of division embedded within negative exponents. Simplification continues with the expansion of \( (3m)^3 \) yielding \( 27m^3 \) and conversion of the term into a fraction. Similarly, \( -2n^{-2} \) simplifies to \( -\frac{2}{n^2} \).

Reciprocal Exponentiation and Its Calculation

Reciprocal exponentiation is the process of raising a number to a negative exponent, creating a reciprocal relationship. This concept is pivotal when working with negative exponents. In algebra, any nonzero number \( a \) raised to a negative exponent \( -b \) is equivalent to the reciprocal of \( a \) raised to the positive exponent \( b \), symbolically shown as \( a^{-b} = \frac{1}{a^b} \).

Using our example \( (3m)^{-3} \) and \( -2n^{-2} \) from the exercise, reciprocal exponentiation allows us to rewrite each term with a negative exponent as a fraction with a positive exponent in the denominator. This transformation simplifies the process of combining terms and finding common denominators if necessary. In practice, the student would rewrite the terms as \( \frac{1}{(3m)^3} \) and \( -\frac{2}{n^2} \) respectively, clearly illustrating that reciprocal exponentiation transforms an expression by 'flipping' the base to the denominator and negating the negative exponent, which makes calculations more straightforward.