Problem 12 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(20^{2}\)

Step-by-Step Solution

Answer

The value of \(20^2\) is 400.

Step 1: Understand the Problem

The problem requires us to determine the value of the power \(20^2\). Here, \(20\) is the base and \(2\) is the exponent which means 20 is to be multiplied by itself.

Step 2: Apply the Exponent Rule

The exponent rule states that when a number is raised to a power, it is multiplied by itself as many times as the exponent indicates. In this case, \(20^2 = 20 \times 20\).

Step 3: Perform the Multiplication

Multiply the base, 20, by itself: \(20 \times 20 = 400\). This is done by multiplying 20 two times because the exponent is 2.

Key Concepts

Powers

The concept of powers is essential in mathematics. It simplifies expressions and represents repeated multiplication succinctly. A power refers to the expression formed when a number, known as the base, is multiplied by itself a number of times specified by the exponent.
For instance, when we write \(20^2\), it means 20 is multiplied by itself twice.
Knowing how to calculate powers helps in understanding and solving many mathematical problems.

• If you have \(a^n\), \(a\) is the base and \(n\) is the exponent.
• The expression means multiply \(a\) by itself \(n\) times.
• Powers help us shorten the writing of repeated multiplication.

Understanding powers allows us to transform complex problems into more manageable calculations.

Base and Exponent

The terms base and exponent are key components of expressions involving powers.

• The base is the number that is raised to a power, meaning it gets multiplied by itself a few times.
• The exponent tells us how many times to use the base in the multiplication.
  For example, in \(20^2\), 20 is the base and 2 is the exponent.
• When the exponent is 2, we call it squaring. If the exponent were 3, it would be cubing.

Understanding which number is the base and which number is the exponent is crucial for calculating powers correctly. It clarifies how many times the base should multiply itself.

Multiplication

Multiplication plays a key role when dealing with powers as it allows us to find the actual value the power represents.
In our example, \(20^2 = 20 \times 20\). The process is straightforward:

• Multiply the base number by itself as many times as the exponent specifies.
• In the case of \(20^2\), you multiply 20 by itself just once more, which equals 400.
• This principle of multiplication is applied to any exponent, whether it is 2 or 10.

Becoming proficient with multiplication helps simplify expressions and solve arithmetic problems that involve powers easily.