Problem 50 - Brief Applied Calculus

Question

Use a calculator to evaluate each expression. Round answers to two decimal places. $$ 5^{0.47} $$

Step-by-Step Solution

Answer

5^{0.47} \approx 2.79

Step 1: Understand the Expression

Our task is to evaluate the expression \( 5^{0.47} \). This involves calculating the value of 5 raised to the power of 0.47.

Step 2: Set Up the Calculation

Using a calculator, you'll enter the base value, which is 5. Next, use the exponentiation key (often depicted as \(^\wedge\) or \(x^y\)) and enter 0.47 as the exponent.

Step 3: Perform the Calculation

Press enter or equals sign on your calculator to compute the value of \(5^{0.47}\). The calculator will process this to find the result.

Step 4: Round the Result

The calculator displays the result as a decimal number. Round this result to two decimal places to complete your computation.

Key Concepts

Calculator Usage

Using a calculator effectively can greatly simplify mathematical problems, especially when dealing with complex equations and exponentiation. To compute something like \( 5^{0.47} \), start by turning on your calculator and ensuring it's in the correct mode for mathematical operations. Most calculators have an exponentiation function, usually labeled as \( x^y \) or a caret symbol (\(^\wedge\)), allowing you to handle powers and roots effortlessly.

Here’s how to perform exponentiation on a calculator:

• Enter the base number, which is 5 in this case.
• Press the exponentiation button (\( x^y \) or \(^\wedge\)).
• Input your exponent, which is 0.47.
• Press the equals button to calculate the result.

This process can handle both whole number and fractional exponents, showcasing the power of scientific calculators.

Rounding Decimals

Rounding decimals is essential in ensuring clarity and preciseness when presenting numerical results, especially in fields such as science, engineering, and finance. After using the calculator to compute \( 5^{0.47} \), the outcome will typically be a number with many decimal places. It’s often necessary to round this to a set number of decimal places to make the figure more understandable.

When the problem specifies to round to two decimal places:

• Look at the third decimal place (the number immediately after the second decimal place).
• If this number is 5 or higher, round the second decimal place up by one.
• If it is less than 5, leave the second decimal place as it is.

Rounding helps bring the focus onto the significant digits, letting you communicate your results effectively.

Mathematical Expressions

Mathematical expressions like \( 5^{0.47} \) represent calculations involving numbers, symbols, and operators to express a particular mathematical idea or operation. Evaluating these expressions calls for understanding not only the meaning of each symbol but also the relationship between them.

In this case:

• The number 5 is the base.
• The number 0.47 is the exponent, indicating the power to which the base is raised.

Exponentiation, as used in this expression, multiplies the base by itself as many times as specified by the exponent. Understanding the components and steps required to evaluate mathematical expressions is crucial in correctly carrying out numerical and algebraic computations.