Question

Simplify, and write without negative exponents. Do some by calculator. $$\left(\frac{2 x}{3 y}\right)^{-2}$$

Short answer

\(\frac{9y^2}{4x^2}\)

Step-by-step solution

Identify Negative Exponent Rule

According to the negative exponent rule, any term with a negative exponent can be rewritten as the reciprocal of that term with a positive exponent. So, a term with a negative exponent like \(a^{-n}\) can be rewritten as \(\frac{1}{a^{n}}\).

Apply Negative Exponent Rule

Apply the negative exponent rule to the given expression \(\left(\frac{2x}{3y}\right)^{-2}\) by taking the reciprocal of the base and changing the exponent to a positive: \(\left(\frac{3y}{2x}\right)^{2}\).

Simplify the Positive Exponent

Simplify the expression with the positive exponent by squaring both the numerator and the denominator separately: \(\frac{(3y)^2}{(2x)^2} = \frac{9y^2}{4x^2}\).

Key concepts

Understanding Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. When an expression like \( b^n \) is presented, it means the base (\( b \)) is multiplied by itself \( n \) times, where \( n \) is the exponent. For instance, \( 3^2 \) indicates that 3 is to be multiplied by itself, resulting in 9.

The rules of exponentiation are not just limited to positive integers. When dealing with negative exponents, the operation handles inversion. This is where the negative exponent rule becomes essential—a powerful tool for simplifying expressions with exponents that are negative.

The Reciprocal of Exponents

The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 5 is \( \frac{1}{5} \). This concept extends to exponents as well. With negative exponents, the rule states that to find the reciprocal of an exponent, you invert the base and make the exponent positive. For example, \( a^{-n} \) becomes \( \frac{1}{a^n} \).

This reciprocal relationship is essential when simplifying expressions because it allows us to express quantities with negative exponents as their positive counterparts, which are often easier to understand and calculate with. Take for instance the expression \( \frac{2x}{3y}^{-2} \); by applying the negative exponent rule, it simplifies to \( \frac{3y}{2x}^2 \), turning a potentially confusing negative exponent into a format most are comfortable with—positive exponents.

Simplifying Expressions with Exponents

Simplifying expressions, especially those with exponents, requires understanding and applying the rules of exponentiation. After applying the reciprocal rule for negative exponents, the next step involves simplifying the expression with its new positive exponent. The expression \( \frac{3y}{2x}^2 \) needs to be approached by squaring both the numerator and the denominator separately. Doing so gives us \( \frac{(3y)^2}{(2x)^2} = \frac{9y^2}{4x^2} \).

This step-by-step method results in an expression that is simpler and more digestible, making it easier not only to understand but also to use in further calculations. Always remember that exponentiation, when handled properly, can simplify and clarify mathematical expressions substantially, enabling students to navigate complex algebraic problems with confidence.