Problem 44 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$\left(\frac{4}{7}\right)^{-0.6}$$

Step-by-Step Solution

Answer

The solution is approximately 1.358705.

Step 1: Understand the Problem

You need to calculate \( \left(\frac{4}{7}\right)^{-0.6} \). This involves taking a fraction, \( \frac{4}{7} \), and raising it to a negative fractional power of \(-0.6\).

Step 2: Apply the Negative Exponent Rule

Recall that \( a^{-b} = \frac{1}{a^b} \). So, \( \left(\frac{4}{7}\right)^{-0.6} = \frac{1}{\left(\frac{4}{7}\right)^{0.6}} \). This means we need to find \( \left(\frac{4}{7}\right)^{0.6} \) first and then take the reciprocal of that value.

Step 3: Calculate \( \left(\frac{4}{7}\right)^{0.6} \)

Use a calculator to compute \( \left(\frac{4}{7}\right)^{0.6} \). Input \( 4 \) divided by \( 7 \) and then raise the result to the power of \( 0.6 \). The calculator should return approximately \( 0.735644 \) on a typical display.

Step 4: Take the Reciprocal

To find \( \left(\frac{4}{7}\right)^{-0.6} \), take the reciprocal of \( 0.735644 \). This means you compute \( \frac{1}{0.735644} \), which yields approximately \( 1.358705 \).

Step 5: Verification

Double-check the calculations with the calculator to ensure accuracy. Confirm that the reciprocal and power steps are correctly executed.

Key Concepts

Negative Exponents

Negative exponents might seem intimidating at first, but they are just the inverse of their positive counterparts. When you see a negative exponent, it indicates that you should take the reciprocal of the number or expression with the positive version of that exponent. For instance:

• If you have a number like \( a^{-b} \), it equals \( \frac{1}{a^b} \).
• Similarly, if you see \( \left(\frac{4}{7}\right)^{-0.6} \), it means you must compute \( \frac{1}{\left(\frac{4}{7}\right)^{0.6}} \).

By understanding negative exponents as a switch to forming reciprocals, you can simplify many problems that might otherwise seem complex. Just remember, flipping to the reciprocal is the key move!
In practice, this concept allows you to solve exercises involving negative powers correctly by focusing first on making the exponent positive and then taking the reciprocal of the result as the last step.

Fractional Exponents

Fractional exponents represent both roots and powers wrapped in a single operation. The most common type you'll see are fractions like \( x^{\frac{m}{n}} \), where a number is both raised to a power and then rooted.

• Think of \( x^{1/2} \) as \( \sqrt{x} \), the square root of \( x \).
• Similarly, \( x^{2/3} \) means \( (x^2)^{1/3} \), or taking the cube root of \( x \) squared.

For the exercise \( \left(\frac{4}{7}\right)^{-0.6} \), the exponent \( 0.6 \) can be seen as a fraction \( \frac{3}{5} \). This means instead of just multiplying, you’re also taking a root. Converting a decimal exponent to a fraction can often give you insights into what the exercise is asking you to do. Decoding these fractional powers helps turn a single operation into a sequence of clearer steps: power and then root. This understanding bridges the gap between intuition and computation, helping you tackle complex exponents with confidence.

Calculator Usage

Calculators are a valuable tool in solving exponentiation problems that involve fractions or difficult decimals. When dealing with powers like \( \left(\frac{4}{7}\right)^{-0.6} \), here’s a step-by-step approach to using your calculator:

• First, divide \( 4 \) by \( 7 \) to get the base value: \(0.5714286\).
• Next, use the power function, often marked as \( y^x \), to raise this result to \( 0.6 \).
• The calculator should then display approximately \( 0.735644 \), which is \( \left(\frac{4}{7}\right)^{0.6} \).
• Finally, take the reciprocal of this result to find \( \left(\frac{4}{7}\right)^{-0.6} \), yielding approximately \( 1.358705 \).

Remember, each calculator might vary slightly in button labels and processes, so it's good practice to familiarize yourself with your specific model. Always double-check your operations to ensure no mistakes occurred during input. Calculators exhibit their full power when handled accurately, particularly with complicated decimal and fractional powers!