Problem 1 - Algebra 2 and Trigonometry

Question

Randy said that \((2)^{3}(5)^{2}=(10)^{5} .\) Do you agree with Randy? Justify your answer.

Step-by-Step Solution

Answer

No, Randy is incorrect because \((2)^{3}(5)^{2}\) equals 200, while \((10)^{5}\) equals 100000.

Step 1: Evaluate the Left-Hand Side

We need to evaluate the expression \((2)^3(5)^2\). First, calculate \((2)^3\):\[(2)^3 = 2 \times 2 \times 2 = 8.\]Next, calculate \((5)^2\):\[(5)^2 = 5 \times 5 = 25.\]Now multiply these results together:\[(2)^3(5)^2 = 8 \times 25 = 200.\]

Step 2: Evaluate the Right-Hand Side

Now, evaluate the expression \((10)^5\).Calculate \((10)^5\):\[(10)^5 = 10 \times 10 \times 10 \times 10 \times 10.\]This can be simplified as:\[10^5 = 100000.\]

Step 3: Compare Both Sides

Now, compare the left-hand side and right-hand side results:- Left-hand side: \(200\)- Right-hand side: \(100000\)Since \(200\) is not equal to \(100000\), Randy's statement is incorrect.

Key Concepts

Base and Exponent

In mathematics, exponentiation is a fundamental operation that involves two key components: the base and the exponent. The base is the number that will be multiplied by itself, and the exponent tells you how many times to perform the multiplication.
You can think of the base as the foundation, and the exponent as the number that dictates the height to which this foundation is built. For example, in the expression \( 5^2 \), "5" is the base, and "2" is the exponent. This means you will multiply 5 by itself, resulting in \( 5 \times 5 = 25 \).

• **Base**: the number that you want to multiply
• **Exponent**: the number of times you multiply the base by itself

This simple yet powerful tool allows you to work with very large numbers in a compact form. Understanding this helps you break down even more complicated expressions easily.

Evaluating Expressions

Evaluating expressions is like solving a mini math puzzle. With exponentiation, first tackle each part by computing the base raised to its exponent. Consider this as executing repeated multiplications of the base.
In our example, the expression \((2)^3(5)^2\) requires you to first solve \((2)^3\) and \((5)^2\) separately.

• Calculate \((2)^3\): \( 2 \times 2 \times 2 = 8 \)
• Calculate \((5)^2\): \( 5 \times 5 = 25 \)

Then multiply the two results: \( 8 \times 25 = 200 \).
Evaluating expressions step-by-step aids comprehension and ensures accuracy, especially when dealing with complex operations. This methodical approach, when applied consistently, helps in solving more complicated mathematical problems effortlessly.

Mathematical Comparison

Mathematical comparison involves analyzing two or more values or expressions to determine relationships like equality or inequality.
In our case, Randy proposed \((2)^3(5)^2=(10)^5\). After evaluating each side, we found that they do not equate. The evaluated results were:\

• *Left-hand side*: \(200\)
• *Right-hand side*: \(100000\)

This illustrates a clear disparity. Since \(200 eq 100000\), Randy's assertion was incorrect.
Comparisons are important in mathematics as they help validate expressions and solutions. They ensure statements made within math are true or need revising, providing a clear understanding of the relationships between different mathematical elements.