Problem 40 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$15^{-1 / 6}$$

Step-by-Step Solution

Answer

The value of \( 15^{-\frac{1}{6}} \) is approximately 0.5946 using a typical calculator.

Step 1: Understand the Expression

The expression we need to solve is \( 15^{-\frac{1}{6}} \). This means we are looking for the reciprocal of the sixth root of 15.

Step 2: Calculate the Sixth Root

Use a calculator to find \( 15^{\frac{1}{6}} \). Enter 15 and then apply the root function, setting the degree to 6, or equivalently raise 15 to the power of \( \frac{1}{6} \).

Step 3: Take the Reciprocal

After finding the sixth root of 15, note the result, and then find its reciprocal. The reciprocal of a number \( x \) is \( \frac{1}{x} \).

Step 4: Use the Calculator for Exact Values

Using a calculator, enter the sixth root value obtained and find its reciprocal to get \( 15^{-\frac{1}{6}} \). Ensure to use all the decimal points your calculator provides.

Step 5: Record the Result

Observe the calculated value, ensuring it contains as many digits as your calculator's display can show.

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of a number by itself. When we see an expression like \( a^n \), it means that the base \( a \) is multiplied by itself \( n \) times. Here, the exponent \( n \) dictates how many times the base number is used as a factor.
For example, \( 3^2 \) means \( 3 \times 3 \), which equals 9. In some cases, the exponent can be a fraction or even a negative number, which introduces more complex concepts like roots and reciprocals.
A fractional exponent like \( a^{\frac{1}{b}} \) means that we take the b-th root of \( a \). In the exercise we have \( 15^{-\frac{1}{6}} \), which we'll break down into two parts: finding a root and then taking the reciprocal.

Reciprocal

A reciprocal of a number is simply 1 divided by that number. Mathematically, the reciprocal of a number \( x \) is expressed as \( \frac{1}{x} \). Multiplying a number by its reciprocal always gives 1.
For instance, the reciprocal of 5 is \( \frac{1}{5} \), and the reciprocal of \( 3/4 \) is \( 4/3 \). In our exercise, we need to find the reciprocal of the sixth root of 15, represented as \( 15^{\frac{1}{6}} \). Once the sixth root is found, taking the reciprocal is straightforward: \( \left(15^{\frac{1}{6}}\right)^{-1} = 15^{-\frac{1}{6}} \).
Reciprocals are especially useful when dealing with division problems or negative exponents, as they allow us to convert these into easier multiplication problems.

Calculator Usage

Using a calculator effectively is crucial for solving problems involving roots and powers, especially when dealing with fractional exponents like \( 15^{-\frac{1}{6}} \). Modern calculators come with functions that make these calculations relatively simple.
Here's a step-by-step way to use a calculator for this specific exercise:

• Enter the base number which is 15.
• If your calculator has a root function, activate it and set the degree to 6 to find \( 15^{\frac{1}{6}} \). Alternatively, raise 15 to the power of \( \frac{1}{6} \) using the exponentiation function.
• After obtaining \( 15^{\frac{1}{6}} \), use the reciprocal button/operation (\( x^{-1} \)) to calculate \( 15^{-\frac{1}{6}} \).
• Make sure your calculator is set to at least display all the digits it possibly can for maximum precision.
• Record the value displayed. This ensures you have as accurate a result as possible, given your calculator's capabilities.

Remember to familiarize yourself with your specific calculator model, as button labels and functions may vary.