Problem 2 - College Algebra

Question

Approximate each number using a calculator. Round your answer to three decimal places. \(3^{2.4}\)

Step-by-Step Solution

Answer

The approximate value of \(3^{2.4}\) rounded to three decimal places is 14.230

Step 1: Power Calculation

Calculate \(3^{2.4}\) using a calculator. When using the calculator, first type in the number '3', followed by '^' which represents the power operation, and then type in the number '2.4', and then equals to get the result.

Step 2: Decimal Rounding

The resulting number will be a decimal. To round this number to three decimal places, look at the fourth decimal place. If this digit is 5 or above, increase the third decimal place by one. Otherwise, keep the third decimal place as it is. Delete all the numbers after the third decimal place to get the rounded number.

Key Concepts

Decimal Rounding

Rounding decimals is an essential skill in mathematics, notably when we require a specific level of precision. This exercise involves rounding to three decimal places. Here's how it's done:

When you have a decimal number and need to round it to three decimal places, follow these steps:

• First, identify the third decimal place. This is the third digit right after the decimal point.
• Next, look at the number directly after the third decimal place, which is the fourth decimal place.
• If this fourth decimal digit is 5 or higher, you will round up the third decimal place by adding one to it.
• If it is less than 5, you will keep the third decimal place unchanged.

For instance, if you have the number 8.6547 and want to round it to three decimal places, you would look at the number 7 (in the fourth place) and round the 8.654 up to 8.655.

Understanding decimal rounding ensures precise calculations and results, which is crucial in subjects like science and engineering.

Calculator Usage

Using a calculator efficiently can significantly ease your math calculations, especially for operations like exponentiation. To perform power operations like in this exercise ('3 raised to the power of 2.4'), you should be comfortable with the calculator's interface.

Here’s a simple way to use a standard scientific calculator for power operations:

• Locate the base number, which in this case is '3', on your calculator.
• Find and press the exponentiation button, typically represented by a symbol like '^' or sometimes 'EXP'.
• After pressing the power button, input the exponent value. Here, you would type '2.4'.
• Press the equals sign to get the result of your calculation.

This calculation will give you a decimal number, which represents the result of raising 3 to the 2.4th power. Remember, scientific calculators might vary slightly in their function keys, but generally, the operation process remains the same. Familiarizing yourself with your calculator model will improve your efficiency and accuracy in solving such exercises.

Power Operation

Power operation, an essential mathematical concept, involves using an exponent to determine how many times to multiply the base by itself. For instance, in the expression \(3^{2.4}\), '3' is the base, and '2.4' signifies the power or exponent.

Understanding this operation is crucial when dealing with exponential functions in various subjects. Here's a clearer breakdown:

• The base is the number that is being multiplied. In above example, it is '3'.
• The exponent indicates how many times the base multiplies itself. A non-whole number like '2.4' suggests a complex form of multiplication, involving roots in most cases.
• For fractional exponents, the computation is akin to finding a specific root alongside multiplying the base by itself.

When we compute \(3^{2.4}\), it implies a mix between multiplying three by itself and taking the root based on '2.4'. This often calls for a calculator to determine the precise value effortlessly.

Mastery of power operations can broaden your understanding in various math and science fields.