Problem 63 - Intermediate Algebra

Question

Use your calculator to evaluate each numerical expression. $$ (1.41)^{4} $$

Step-by-Step Solution

Answer

The value of \((1.41)^{4}\) is approximately 3.9493.

Step 1: Understanding the Expression

We are asked to evaluate the numerical expression \((1.41)^{4}\). This means we need to multiply 1.41 by itself four times.

Step 2: Initial Calculation

Start by calculating \(1.41^2\). Multiply 1.41 by itself: \(1.41 \times 1.41 = 1.9881\).

Step 3: Multiply Again

Now we need to use the result from Step 2 and multiply it by 1.41 again to find \(1.41^3\). Calculate \(1.9881 \times 1.41 = 2.8052\).

Step 4: Final Multiplication

Multiply the result from Step 3 by 1.41 one more time to find \(1.41^4\). Calculate \(2.8052 \times 1.41 = 3.949332\).

Step 5: Rounding the Result

Round the final answer to four decimal places, if necessary. In this case, the answer is already given to a suitable accuracy of four decimal places: 3.9493.

Key Concepts

Numerical Expression

In mathematics, a numerical expression is a combination of numbers and operations like addition, subtraction, multiplication, and division, without any equal signs. These expressions do not resolve to a single definite value until all operations are completed. In the given exercise, the numerical expression \((1.41)^4\) involves an operation called exponentiation, where 1.41 is the base and 4 is the exponent. This means multiplying 1.41 by itself four times.

• Step 1: Calculate \(1.41^2\) by doing \(1.41 \times 1.41 = 1.9881\).
• Step 2: Use the result to find \(1.41^3\) with \(1.9881 \times 1.41 = 2.8052\).
• Step 3: Finally, calculate \(1.41^4\) by multiplying \(2.8052 \times 1.41 = 3.949332\).

Understanding how to break down and solve numerical expressions is key to mastering many mathematical problems.

Rounding Decimals

Rounding decimals is the process of adjusting a decimal number to a nearby, cleaner value, often with fewer decimal places. In everyday math, rounding helps simplify numbers. It is especially useful in making lengthy calculations more manageable.
For the expression \((1.41)^4\) in our exercise, we obtained the result 3.949332. To present this in a more simplified form, we round it to four decimal places. Here's a quick guide on rounding:

• Look at the fifth decimal place: If it is 5 or more, round up the fourth decimal place by 1.
• If it is less than 5, leave the fourth decimal as it is.

So, in the exercise, 3.94933 is rounded to 3.9493. Understanding rounding can help maintain precision in calculations while simplifying expressions.

Calculator Usage

Using a calculator correctly can greatly enhance your efficiency in solving mathematical problems, especially when dealing with complex operations like exponentiation. Calculators compute numerical expressions quickly, reducing human error and saving time.

• Start by ensuring that your calculator is in the correct mode (degree/radian or standard mode).
• For this exercise, input \(1.41^4\) directly into the calculator for a quick result. You simply press '1.41', then the exponent function button, and finally '4'.
• Most calculators will immediately offer the result, which should match 3.949332.

While calculators are powerful tools, understanding the mathematical steps behind the operations is crucial for verifying your results and building strong numerical problem-solving skills.