Problem 8 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(11^{2}\)

Step-by-Step Solution

Answer

The value of \(11^2\) is 121.

Step 1: Understand the Power Notation

The problem asks us to determine the value of the power \(11^2\). A power consists of a base number and an exponent. The base is 11, and the exponent is 2, indicating how many times to multiply the base by itself.

Step 2: Perform the Multiplication

To solve \(11^2\), you multiply the base, 11, by itself. This operation is: \(11 \times 11\).

Step 3: Calculate the Result

Calculate the multiplication: \(11 \times 11 = 121\). Thus, the value of \(11^2\) is 121.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. For example, in the expression \(11^2\), the number 11 is the base, and the number 2 is the exponent. The exponent tells you how many times the base is used as a factor in multiplication. Therefore, \(11^2\) means multiplying 11 by itself once (since the exponent is 2). This concept is essential in simplifying expressions and solving equations that involve repeated multiplication.

• Exponential growth: Exponents represent significant growth in numbers, as each increase in the exponent multiplies the base by itself again.
• Powers of ten: A common application of exponents is in powers of ten, like \(10^3 = 1000\).

Understanding exponentiation can simplify complex mathematical problems into manageable steps.

Multiplication

Multiplication is one of the fundamental arithmetic operations. It combines equal groups of things into a larger group and is often seen as repeated addition. For example, multiplying \(11 \times 11\) means adding the number 11 to itself 11 times. Multiplication is associative and commutative, meaning the order of multiplication does not affect the result - both \(11 \times 2\) and \(2 \times 11\) will yield 22.

• Associative property: Changing the grouping of numbers, such as \( (11 \times 2) \times 3 \), does not change the product.
• Commutative property: Order doesn't matter, so \(11 \times 3\) is the same as \(3 \times 11\).

By understanding this operation, complex problems involving multiplication can be broken down into simpler parts.

Base and Exponent

The base in exponentiation is the number that is being multiplied. The exponent tells you how many times to use the base as a multiplier. In the expression \(11^2\), 11 is the base, and 2 is the exponent. This means you multiply 11 by itself for one less than the exponent. Using the wrong base or exponent changes the entire calculation. Let's explore:

• If you have \(11^3\), you calculate \(11 \times 11 \times 11\).
• A base of 1 with any power, \(1^n\), is always 1 because 1 multiplied by itself any number of times is always 1.

Understanding the role of base and exponent is crucial in mathematics to ensure accurate calculations in both practical and theoretical applications.