Problem 34 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(11^{2}\)

Step-by-Step Solution

Answer

\(11^2 = 121\).

Step 1: Understand the Power

The power notation \(11^2\) means 11 raised to the power of 2, which is the same as multiplying 11 by itself.

Step 2: Multiply the Base by Itself

To calculate \(11^2\), you multiply 11 by 11: \(11 \times 11\).

Step 3: Perform the Multiplication

Now, do the multiplication: \(11 \times 11 = 121\).

Step 4: Use a Calculator to Check

Confirm your answer with a calculator. Enter 11, press the power button, enter 2, and you will get 121.

Key Concepts

Calculating Powers

Calculating powers involves raising a number, known as the base, to an exponent. In the exercise we've looked at, the base is 11 and the exponent is 2, written as \(11^2\). This notation indicates that you need to multiply the base (11) by itself for the number of times indicated by the exponent (2).

In simpler terms, \(11^2\) asks you to compute \(11 \times 11\), which equals 121. Calculating powers is a fundamental skill in mathematics, helping to simplify expressions and solve equations involving repeated multiplication easily.

Using a Calculator for Exponents

Using a calculator to solve exponent problems can simplify the process and ensure accuracy, especially when dealing with larger numbers. Most scientific calculators have a specific function for exponents. Here's a simple guide on how to use it:

• First, enter the base number, which is 11 in our exercise.
• Next, find and press the "power" or "exponent" button, often marked as \(^\) or \(x^y\).
• Finally, enter the exponent, which is 2 in this case.
• Press the "equals" button to see the result.

For \(11^2\), the calculator should display 121. The advantage of using a calculator lies in its ability to handle complex calculations quickly, minimizing errors, especially in more advanced tasks.

Multiplication

Multiplication is the mathematical operation of scaling one number by another. At its most basic, it involves adding a number to itself a certain number of times. When calculating powers like \(11^2\), you perform multiplication by repeated addition.

In \(11 \times 11\), you are combining the number 11 with itself. The multiplication process is straightforward:

• Write the two numbers (the multiplicands) one above the other.
• Starting from the right, multiply each digit of the lower number by each digit of the upper number.
• Sum the resulting products to get the final result.

Here the simple multiplication yields 121, showing the result of \(11^2\). Multiplication is a cornerstone of arithmetic operations and is essential for understanding powers and more complex algebraic functions.