Problem 69 - Elementary Algebra

Question

\((3 x)^{-2}\)

Step-by-Step Solution

Answer

\((3x)^{-2} = \frac{1}{9x^2}\)

Step 1: Understand the Negative Exponent

The expression \( (3x)^{-2} \) contains a negative exponent. A negative exponent indicates that the base should be taken as the reciprocal and the exponent changed to positive. In this case, the base is \( 3x \).

Step 2: Apply the Negative Exponent Rule

Rewrite the expression \( (3x)^{-2} \) by taking the reciprocal of \( 3x \) and changing the exponent to positive: \[\frac{1}{(3x)^2}\]

Step 3: Simplify the Denominator

Now calculate \((3x)^2\). The exponent applies to both the coefficient and the variable: \[(3x)^2 = 3^2 \times x^2 = 9x^2\]

Step 4: Final Expression

Substitute back into the expression using the simplified denominator: \[\frac{1}{9x^2}\]

Key Concepts

exponent rules

Exponent rules are guidelines that help simplify and calculate expressions involving exponents. One key rule involves negative exponents. A negative exponent indicates taking the reciprocal of the base and then raising it to the corresponding positive exponent. For example, \((3x)^{-2}\) becomes \[ \frac{1}{(3x)^2} \]. There are other important exponent rules:
- A product of powers rule: \(a^m \times a^n = a^{m+n}\).
- A power of a power rule: \( (a^m)^n = a^{m \times n} \).
- A power of a product rule: \( (ab)^n = a^n \times b^n \).
Understanding these rules is crucial for simplifying expressions efficiently.

reciprocal

The term 'reciprocal' means flipping a fraction. For any non-zero number \(a\), the reciprocal is \( \frac{1}{a} \). When dealing with negative exponents, finding the reciprocal is essential. The expression \( (3x)^{-2} \) transforms into its reciprocal, then applies the positive exponent: \[ \frac{1}{(3x)^2} \]. The concept of a reciprocal extends to variables and more complex expressions. If \(y = 3x\), then its reciprocal is \[ \frac{1}{3x} \]. Simplifying expressions becomes easier once the reciprocal process is understood.

simplifying expressions

Simplifying expressions involves transforming them into a simpler or more manageable form. It often includes using exponent rules and finding reciprocals. Consider our example: \( (3x)^{-2} \). We begin by writing it as \[ \frac{1}{(3x)^2} \]. Next, calculate \( (3x)^2 = 3^2 \times x^2 \). This results in \( 9x^2 \). So the simplified expression is \[ \frac{1}{9x^2} \]. The goal of simplifying expressions is to make them more straightforward to work with, whether for further calculation or understanding their properties.

algebra

Algebra is a branch of mathematics dealing with symbols and the rules to manipulate those symbols. In algebra, we often encounter variables and exponents, like in the expression \( (3x)^{-2} \). To solve such problems, we apply algebraic rules like the exponent rules. Here, we recognize the negative exponent and transform the expression into \[ \frac{1}{(3x)^2} \]. Algebra involves combining like terms, solving equations, and simplifying expressions. These techniques are fundamental to solving more complex mathematical problems and understanding relationships between quantities.