Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$\left(\frac{5}{6}\right)^{-1.3}$$

Step-by-Step Solution

Answer

The result is approximately 1.247.

Step 1: Understanding the Problem

We need to evaluate the expression \(\left(\frac{5}{6}\right)^{-1.3}\). This involves raising the fraction \(\frac{5}{6}\) to the power of \(-1.3\).

Step 2: Changing the Negative Exponent

Recall that raising a number to a negative power involves taking the reciprocal of the base and then raising it to the positive of that power. Therefore, \(\left(\frac{5}{6}\right)^{-1.3} = \left(\frac{6}{5}\right)^{1.3}\).

Step 3: Calculating the Power

Use a calculator to find \(\left(\frac{6}{5}\right)^{1.3}\). First, compute \(\frac{6}{5} = 1.2\). Then, use the calculator to find \(1.2^{1.3}\).

Step 4: Input into Calculator

Enter \(1.2\) and then use the exponent function to raise it to \(1.3\). Depending on the calculator, this might involve using buttons like '^', 'power', or similar.

Step 5: Reading the Display

After entering the expression into the calculator, the display shows the result. For \(1.2^{1.3}\), the calculator might show a result such as \(1.247\). (The exact number will depend on the calculator's precision.)

Key Concepts

Negative Exponents

When you see a negative exponent, don't let it intimidate you. It simply means you're working with the reciprocal of the base. For example, if you have \(a^{-n}\), this is the same as \(\frac{1}{a^n}\). In simple terms, flip the base (turn it upside down) and remove the negative in the exponent. In our exercise, this meant turning \(\left(\frac{5}{6}\right)^{-1.3}\) into \(\left(\frac{6}{5}\right)^{1.3}\). This approach simplifies the calculation and turns a negative exponent into a positive one, making it easier to work with and eventually evaluate using a calculator.

Fractional Exponents

Fractional exponents are an elegant way to express roots and powers in a compact form. For instance, \(a^{1/n}\) indicates the \(n\)-th root of \(a\). In the case of more complex expressions, such as \(a^{m/n}\), it combines both rooting and powering. It can be understood as \((a^{1/n})^m\), which means first taking the root and then raising the result to a power. In our example, we computed \(1.2^{1.3}\), which involves both root and power operations done in sequence. By using fractional exponents, calculations become streamlined and often more manageable, especially when used in advanced mathematical operations.

Calculator Use

When using calculators for evaluating powers and roots, a few steps can make the process smooth. Start by locating and familiarizing yourself with the exponent function. It might be labeled as '^', 'power', or similar buttons. After identifying this, enter the base number, followed by the exponent symbol, and then input the exponent value. For our example, once you enter \(1.2\), using the power function to input \(1.3\) results in the correct calculation of \(1.2^{1.3}\). Make sure to carefully follow each button press and type the numbers precisely to ensure accuracy. Depending on your calculator's precision, results might slightly vary, but typically, modern calculators will show several decimal places for precision in your displayed solution.