Problem 14 - A Graphical Approach to College Algebra

Question

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$6.4^{-\sqrt{3}}$$

Step-by-Step Solution

Answer

The approximation is about 0.062716699.

Step 1: Understanding the Expression

The expression given is \(6.4^{-\sqrt{3}}\). The base here is 6.4, and the exponent is the negative square root of 3. We need to evaluate this using a calculator.

Step 2: Evaluating the Square Root

Using a calculator, find the value of \(\sqrt{3}\). This is approximately 1.73205081 when displayed to the maximum decimal places on most calculators.

Step 3: Applying the Negative Exponent

Now, since the exponent is negative, we need to calculate \(6.4^{1.73205081}\) and then take the reciprocal of that value. For negative exponents, \(a^{-b} = \frac{1}{a^b}\).

Step 4: Calculating the Power

Find \(6.4^{1.73205081}\) using the calculator. It should give a large value since we are raising the base to a fractional power.

Step 5: Finding the Reciprocal

Take the reciprocal of the result from Step 4 to complete the calculation: \(6.4^{-\sqrt{3}} = \frac{1}{6.4^{1.73205081}}\).

Step 6: Displaying the Final Result

Input the values in a scientific calculator, which provides the result with maximum displayed decimal places. The final approximation for \(6.4^{-\sqrt{3}}\) is approximately 0.062716699.

Key Concepts

Understanding Negative Exponents

In mathematics, a negative exponent represents an operation where instead of multiplying, you perform division. When you see something like \( a^{-b} \), it means you take the number \( a \) to the power of \( b \), and then take the reciprocal of that result. Therefore, \( a^{-b} = \frac{1}{a^b} \). Negative exponents can sometimes seem confusing, because when the exponent is positive, we think of repeated multiplication, but a negative exponent flips that thought to repeated division.
For instance, if you have \( 2^{-3} \), it doesn't mean the cube root of 2. Instead, it translates to \( \frac{1}{2^3} \), which equals \( \frac{1}{8} \) or 0.125. This understanding is crucial when dealing with mathematical expressions that include negative exponents, such as in the given exercise, where the exponent \(-\sqrt{3}\) means you need to find \( 6.4^{\sqrt{3}} \) first and then take the reciprocal of that value.

Exploring Square Roots

A square root represents a value that, when multiplied by itself, gives the original number. The symbol \( \sqrt{} \) is used to denote a square root. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). In our exercise, we deal with \( \sqrt{3} \), a non-perfect square. Many square roots, like \( \sqrt{3} \), are irrational numbers, meaning they cannot be expressed exactly as a simple fraction and have non-repeating, non-terminating decimal parts.

• This value, \( \sqrt{3} \), approximates to 1.73205081 when displayed to the maximum number of decimal places on many calculators.
• Knowing how to approximate such values is extremely helpful when you need an exact decimal to input into calculations, especially with negative exponents or other complex operations.

By understanding square roots more deeply, you become capable of handling them in diverse mathematical scenarios, ensuring you have the right values for further calculations.

Using Calculators Effectively

Calculators are powerful tools for solving complex mathematical problems. Knowing how to use them effectively can save you time and improve accuracy. In our exercise, we leverage a calculator to:

• Find square roots, like \( \sqrt{3} \), enabling a clear decimal representation.
• Compute powers with fractions or negative exponents, such as \( 6.4^{\sqrt{3}} \), which involves entering the values precisely into the calculator.
• Automatically compute reciprocals when dealing with negative exponents, by recognizing patterns like \( a^{-b} = \frac{1}{a^b} \).

When using calculators, ensure:
Your inputs are exact and avoid rounding until the end to maintain precision
The calculator is in the correct mode (for example, degree vs. radian mode in trigonometry) to avoid unintended errors.
Understanding the full functionalities of your calculator, including how to use memory operations and advanced functions, can further enhance your math-solving capabilities.