Problem 1 - College Algebra

Question

In Exercises \(1-10,\) approximate each number using a calculator. Round your answer to three decimal places. $$ 2^{3.4} $$

Step-by-Step Solution

Answer

The solution to the math problem \(2^{3.4}\) rounded to three decimal places is 10.370.

Step 1: Compute the exponential expression

Calculate the value of \(2^{3.4}\) using a calculator or a scientific computation software.

Step 2: Round your answer

Next, round this result to three decimal places. A reminder of how to round numbers: If the fourth decimal place is a digit from 5 to 9 increment the third place by one, if it's a digit from 0 to 4, leave the third digit unchanged.

Key Concepts

Exponents

Understanding exponents is crucial to working with exponential functions and expressions. An exponent represents the number of times a number, known as the base, is used as a factor in a multiplication.
For instance, in the expression \(2^{3.4}\), \(2\) is the base and \(3.4\) is the exponent. This means you are multiplying \(2\) by itself \(3.4\) times, however, being a non-integer exponent, it entails more complex operations than simple repeated multiplication.

• For integer exponents, like \(2^3\), it simply involves multiplying \(2\times2\times2\).
• For fractional exponents, it incorporates both power and root operations.
• For decimals, calculators or software come in handy as they use logarithmic functions to compute them.

Grasping this concept helps streamline calculations involving exponentially growing or decaying processes, as seen in finance, physics, or population studies.

Rounding Numbers

Rounding numbers is a technique used to reduce the number of decimal places. This is particularly useful when dealing with long decimal figures that do not significantly impact the precision of the solution.
When rounding to three decimal places, following is key:

• Identify the third decimal place.
• Look at the digit immediately to its right, also known as the fourth digit.
• If the fourth digit is 5 or greater, increase the third digit by one.
• If it is 4 or less, keep the third digit as is.

For example, rounding \(9.87654\) to three decimals would result in \(9.877\), as the fourth digit, \(5\), indicates increasing the third digit. This method keeps your answers both precise and manageable.

Calculator Usage

Using a calculator efficiently can simplify computing exponential expressions. Calculators come programmed with functions that enable quick calculations, leading to accurate results without manual error.
Here is a simple guide to using calculators for exponentiation:

• Identify and access the exponent function, often represented as "^" or "exp" on the keypad.
• Input the base number first.
• Enter the exponent using the exponent button.
• Press equals or enter to obtain the result.

Make sure your calculator is in the correct mode (degree or radian) when dealing with trigonometric functions alongside exponentials. This makes tasks with non-integer exponents straightforward, as seen with computing \(2^{3.4}\). Calculators ensure efficiency and precision in tackling mathematical exercises.