Problem 60 - Algebra 2

Question

State whether each statement is true or false when \(n=1\). Explain. \(3^{n}-1\) is even

Step-by-Step Solution

Answer

True; when \(n=1\), \(3^n - 1 = 2\) which is even.

Step 1: Substitute the value of n

Substitute the given value of \(n = 1\) into the expression \(3^n - 1\). This will convert the expression into numeric form.

Step 2: Simplify the Expression

Calculate the expression \(3^1 - 1\). This involves computing the power and then the subtraction.

Step 3: Calculate the Power

The power of 3 to the 1st power is 3, because raising any number to 1 just returns itself. So, we have \(3^1 = 3\).

Step 4: Perform the Subtraction

Subtract 1 from 3: \(3 - 1 = 2\). This simplifies the expression to 2.

Step 5: Check Evenness

Determine if the result, 2, is an even number. An even number is divisible by 2 without any remainder.

Step 6: Conclusion

Since 2 is divisible by 2, it is even. Therefore, the statement \(3^n - 1\) is even when \(n=1\) is true.

Key Concepts

Understanding Integer Exponents

Integer exponents refer to the process of multiplying a number, called the base, by itself a certain number of times. When an exponent is positive, it tells us how many times to use the base as a factor. The notation used is a small number written above and to the right of the base number, like this: \(a^n\), where \(a\) is the base and \(n\) is the exponent.
For example, \(3^1\) means multiply 3 by itself 1 time, which simply returns 3. This is a straightforward concept, yet it's fundamental when dealing with more complex expressions.
When \(n=1\), any base raised to the first power remains unchanged, because the exponent 1 implies the base itself is the only factor. Thus, \(3^1 = 3\). This understanding simplifies the computation process in expressions like \(3^n - 1\).

• The base remains the same after raising it to the power of 1.
• The result is always the base itself when the exponent is 1.

This principle is crucial when simplifying mathematical expressions with integer exponents.

Simplifying with Subtraction

Subtraction is one of the basic arithmetic operations where we find the difference between two numbers; this means taking one number away from another. In mathematical expressions, subtraction is symbolized by the minus sign \(-\). For example, in \(3-1\), the 1 is being subtracted from 3.
The result of subtraction forms part of understanding expressions when combined with other operations, such as exponentiation. In our example, after calculating \(3^1\), we move on to subtract 1 from 3. This gives us:

• Perform the subtraction \(3-1\).
• Find that the answer is 2.

Subtraction simplifies the expression to a single number once all steps are combined. This procedure helps in checking further properties of the expression, like identified in the exercise solution.

Exploring Number Properties

Number properties are essential for understanding various characteristics and behaviors of numbers, especially regarding even and odd numbers. An even number is any integer that can be divided by 2 with no remainder, typically ending in 0, 2, 4, 6, or 8. For example, the number 2 is even because \(2 \div 2 = 1\) without a remainder.
Addition, subtraction, multiplication, and division of numbers follow these intrinsic properties, helping determine the evenness or oddness of results. In the expression \(3^n - 1\), simplifying it to \(2\) leads us to use the knowledge of number properties to say:

• If a number ends in 0, 2, 4, 6, or 8, it is even.
• Dividing an even number by 2 leaves no remainder.

Thus, understanding these properties ensures that we can reliably state whether outcomes of expressions like \(3^n - 1\) at given values of \(n\) are even or not. Using these properties helps affirm the truth in mathematical statements, ensuring precise and dependable conclusions.