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Chapter 1: Problem 1

Decide whether each statement is true \(o r\) false. If it is false, explain why. $$ 3^{2}=6 $$

Short Answer

Expert verified
False; \(3^2\) equals \(9\), not \(6\).

Step by step solution

01

Calculate the Left Side of the Equation

Calculate the left-hand side of the equation: Evaluate the exponentiation: \(3^2\) gives us \(3\times3=9\)
02

Compare Both Sides of the Equation

Compare the result from the previous step to the right-hand side of the equation: The left side is \(9\), and the right side is \(6\). Since \(9eq6\), the equation \(3^2=6\) is false.
03

Conclude the Truth Value

Based on the comparison in Step 2, conclude whether the statement is true or false:The statement is false because the left-hand side (\(9\)) does not equal the right-hand side (\(6\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponentiation
Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power or exponent. The exponent indicates how many times the base is multiplied by itself. For example, in the expression \(3^2\):
- The base is 3.
- The exponent is 2, which tells us to multiply 3 by itself once.
So, \(3^2\) means \(3 \times 3\), which equals 9.
It's important to perform exponentiation correctly to solve algebraic equations accurately.
Equation Comparison
Comparing equations is a critical step in algebra. It means evaluating both sides of an equation to see if they are equal. To determine whether an algebraic statement is true or false, like in our exercise, we:
- Calculate the value on each side of the equation separately.
- Compare these results.
For example, with the equation \(3^2 = 6\), we:
1. Calculate the left side: \(3^2 = 9\).
2. Compare it to the right side: 6.
We see that 9 does not equal 6, so the equation is false.
Truth Value
In mathematics, determining the truth value of an equation involves establishing whether the statement it makes is true or false. When evaluating truth values:
- Calculate both sides of the equation.
- Compare the results.
If both sides are equal, the statement is true; otherwise, it's false.
In our example, the given equation \(3^2 = 6\) was evaluated:
- The left side calculates to 9.
- The right side is given as 6.
Because 9 does not equal 6, the truth value of the statement is false.

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