Problem 3 - College Algebra Essentials

Question

Approximate each number using a calculator. Round your answer to three decimal places. $$3^{\sqrt{5}}$$

Step-by-Step Solution

Answer

The approximate value of \(3^{\sqrt{5}}\) is 12.669

Step 1: Determine the square root of 5

Use your calculator to find the square root of 5. Always remember to check your calculator's settings to make sure it's in the correct mode.

Step 2: Compute 3 to the power of square root of 5

Raise 3 to the power of the result you got in step 1. This can be done using the exponent key on your calculator.

Step 3: Round off the answer

Round off your answer to three decimal places. This involves looking at the fourth decimal place and if it's 5 or greater, the third decimal place is increased by 1, if it's less than 5, the third decimal place stays the same.

Key Concepts

Square Root

The square root is a mathematical function that helps us determine a number which, when multiplied by itself, gives us the original number. For instance, the square root of 9 is 3, since \(3 \times 3 = 9\). In this exercise, we're dealing with \(3^{\sqrt{5}}\). This means we need to first find the square root of 5 before we proceed.
To calculate \(\sqrt{5}\), we use a calculator since it's not a perfect square. For many students, imagining the square root might be tricky. Simply think of it as asking: "What number multiplied by itself is nearly 5?" Your calculator will provide this answer quickly. Make sure your device is set correctly to give you a decimal output, and type in \(\sqrt{5}\) to find the value, which is approximately 2.236.

Calculator Use

Calculators can be a student's best friend when dealing with complex numbers and tasks like this one. When using a calculator to solve \(3^{\sqrt{5}}\), start by finding the square root of 5, as we've established earlier. Understanding how to effectively use your calculator is vital:

• Ensure it's in the right mode, usually 'normal' mode for these calculations.
• Locate the square root button, often denoted as \(\sqrt{}\) or partially hidden under a secondary key function.
• Use the exponent key (usually labeled as \(x^y\) or similar) to raise 3 to the power of the result from your earlier calculation.

With these actions, your calculator becomes an essential tool for breaking down complex math problems into manageable steps.

Rounding Off

Rounding off is a technique that helps simplify a number while maintaining its value close to the original. You might need to round your final result to three decimal places for precision's sake.
Here's how you do it effectively:

• Look at the number immediately after your third decimal place.


• If this number is 5 or higher, increase the third decimal by one.


• If it's less than 5, leave the third decimal as it is.

In our case, to round the result of \(3^{\sqrt{5}}\), which might be something like 15.58892, you look at the fourth number (8) after the decimal point. Since this number is more than 5, you increase the third decimal place, rounding your number to 15.589.