Problem 17 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by hand. $$ \left(\frac{2}{3}\right)^{-2} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{9}{4} \).

Step 1: Understand the Negative Exponent

When an expression has a negative exponent, you can rewrite it as the reciprocal of the base raised to the opposite positive exponent. For the expression \( \left(\frac{2}{3}\right)^{-2} \), this means we take the reciprocal of \( \frac{2}{3} \) and raise it to the positive power. So, \( \left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 \).

Step 2: Apply the Power to Both Numerator and Denominator

Apply the exponent of 2 to both the numerator and the denominator of the fraction \( \frac{3}{2} \). This results in \( \frac{3^2}{2^2} \).

Step 3: Calculate the Powers

Calculate \( 3^2 \) and \( 2^2 \). \( 3^2 = 9 \) and \( 2^2 = 4 \), so \( \frac{3^2}{2^2} = \frac{9}{4} \).

Key Concepts

Reciprocal of a Fraction

Understanding the concept of a reciprocal is essential, especially when dealing with negative exponents. A reciprocal is simply the inverse of a given fraction. To find the reciprocal of a fraction like \( \frac{2}{3} \), you simply swap the numerator and the denominator. Thus, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
You can think of it as turning the fraction "upside down." This process becomes particularly useful when you encounter expressions with negative exponents. Since a negative exponent indicates the reciprocal of the base, turning the fraction over is precisely what you need to do.
For example, in the expression \( \left(\frac{2}{3}\right)^{-2} \), the negative exponent tells you to use the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). Then, the problem becomes more manageable because you simply use positive exponents from there.

Exponentiation in Algebra

Exponentiation in algebra is a fundamental operation that involves raising a number to a power. When given a base and an exponent, the exponent indicates how many times you should multiply the base by itself.
This concept extends stress-free to both whole numbers and fractions.
For example, considering the expression \( \left(\frac{3}{2}\right)^2 \), you apply the exponent of 2 to both the numerator and the denominator. This means you calculate \( 3^2 \) for the numerator and \( 2^2 \) for the denominator.
Moreover, you'll express them as: \( 3 \times 3 = 9 \) and \( 2 \times 2 = 4 \), resulting in the fraction \( \frac{9}{4} \).
It’s that simple! Remember, exponentiation is a way to perform repeated multiplication succinctly.

Fractional Exponents

Fractional exponents can initially seem intimidating, but they simplify many algebraic manipulations! A fractional exponent represents both a power and a root. It’s commonly expressed as \( a^{m/n} \), which means you take the \( n \)-th root of \( a \) and then raise it to the \( m \)-th power, or vice versa.
While our main example doesn't directly address fractional exponents, understanding this translates to deeper insights when you encounter them in similar exercises. For instance, if you have the expression \((x^2)^{1/2}\), it symbolizes the square root of \( x^2 \), simplifying famously to \( x \).
This dual interpretation—combining powers and roots—is potent for solving whiz-level algebra problems and comes in handy often. Refreshing your skills on fractional exponents paves the way for more advanced mathematical challenges with confidence.