Problem 5 - College Algebra

Question

In Exercises \(1-10,\) approximate each number using a calculator. Round your answer to three decimal places. $$ 4^{-1.5} $$

Step-by-Step Solution

Answer

By following the steps correctly, you are expected to find that \(4^{-1.5} \approx 0.189

Step 1: Understand the Problem

The goal is to use a calculator in order to compute an approximation of \(4^{-1.5}\), and to round the result to three decimal places.

Step 2: Input Expression into Calculator

Type the expression \(4^{-1.5}\) into your calculator. Make sure to correctly use the negative exponent.

Step 3: Approximate and round the result

Observe the result. The calculator should display a series of digits. Take note of the first three digits after the decimal point and then look at the fourth digit. If the fourth digit is 5 or above, you need to round up the third digit. If the fourth digit is 4 or below, you leave the third digit as it is.

Key Concepts

Negative Exponents

Understanding negative exponents is crucial when dealing with exponential expressions. A negative exponent indicates that the base, which is the number being raised to a power, should be taken as the reciprocal and then raised to the absolute value of that exponent. In simple terms, for any non-zero number 'a' and any positive integer 'n', the expression
\(a^{-n} = \frac{1}{a^{n}}\)
holds true. So, when you encounter an expression like \(4^{-1.5}\), it's the same as calculating \(\frac{1}{4^{1.5}}\). Calculators typically have a dedicated button for entering exponents, and it is important to enter the negative sign correctly to ensure an accurate result.

Rounding Decimals

Rounding decimals is a method used to shorten a number to a certain number of decimal places without significantly changing its value. This is particularly useful when the exact decimal is either unknown or unnecessary. The process is straightforward: if you want to round to three decimal places, identify the fourth decimal place. If this digit is 5 or greater, increase the third decimal place by one; if it is 4 or lower, the third decimal place stays the same. For instance, if the calculator shows \(0.12345\), rounded to three decimal places it becomes \(0.123\). If it were \(0.12355\), it would round up to \(0.124\). Accurate rounding ensures that the number remains a close approximation to the original value.

Calculator Usage

Entering Expressions

Using a calculator effectively requires familiarizing oneself with its functions. For exponential expressions specifically, most calculators have a button labeled with a caret (^) or displaying an exponent 'e'; this is used to enter the exponent. With negative exponents, ensure you use the negative sign, often separate from the subtraction key.

Estimating Results

When you approximate an expression like \(4^{-1.5}\), the calculator provides a decimal result. It's vital to interpret this output correctly. Calculators may differ in the way they display large or small numbers; some use scientific notation for very large or small results, which you should be able to switch back to a standard decimal format for ease of rounding.

Exponential Expressions

Exponential expressions represent repeated multiplication of a base number. An expression like \(a^n\) tells you to multiply 'a' by itself 'n' times. When dealing with fractional exponents, the numerator indicates the power to raise the base to, while the denominator indicates the root. For instance, \(4^{1.5}\) is the same as \(4^\frac{3}{2}\), which means the square root of \(4^3\), or \(\sqrt{64} = 8\). It's essential when finding these values, especially with negative or fractional exponents, to correctly enter the base and exponent into your calculator to ensure accurate results. Understanding these expressions and how to properly handle them using a calculator is a valuable skill in both academic and real-world contexts.