Problem 88 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. (Lesson 2.5) $$2(-1)(-x)^{3} \text { when } x=-3$$

Step-by-Step Solution

Answer

The evaluated expression for \(x = -3\) in the equation \(2(-1)(-x)^{3}\) is \(-54\).

Step 1: Understand the Expression

The expression is \(2(-1)(-x)^{3}\), and it is given that \(x = -3\). The first thing to do is to understand what this expression is instructing to do. Firstly, there is multiplication between two numbers and a variable raised to an exponent. Remember that a variable with a negative sign raised to an odd exponent will result in a negative value. Therefore, \((-3)^{3}\) becomes \(-27\). Moreover, \(-1\) multiplied by any number maintains the sign of that number.

Step 2: Substitute the Value

Substitute the given \(x = -3\) into the equation, which now becomes: \(2(-1)(--3)^{3}\). Keep in mind that double negatives equal a positive.

Step 3: Simplify Expression Post-Substitution

Now simplify the expression. First, observe that when 3 is raised to the exponent of 3: \(3^3 = 27\). Therefore, the expression now becomes \(2(-1)(27)\).

Step 4: Complete multiplication

Finally, perform the multiplications. \(2(-1) = -2\) and this makes our expression \(-2 * 27\) which equals \(-54\).

Key Concepts

Exponents

Understanding exponents is key in algebra. An exponent tells us how many times a number, known as the base, is used as a factor. For example, in the expression \((-3)^3\),

• \(-3\) is the base.
• 3 is the exponent, meaning you multiply \(-3\) by itself three times: \((-3) \times (-3) \times (-3)\).

When the base is negative and the exponent is odd, the result is negative. This is because every pair of negatives multiplies to a positive, but an extra negative keeps it negative. So, \((-3)^3 = -27\).
Remember, if the exponent were even, the result would be positive, since all negative signs would pair off.

Multiplication of Integers

Multiplying integers follows specific rules that are easy to remember. If you multiply two positive numbers, the result is positive.
Similarly, multiplying two negative numbers also produces a positive result due to the double negation.

• Positive \( \times \) Positive = Positive
• Negative \( \times \) Negative = Positive
• Positive \( \times \) Negative = Negative
• Negative \( \times \) Positive = Negative

In the expression \(2(-1)(27)\),

• First, multiply \(2\) by \(-1\) to get \(-2\).
• Then multiply \(-2\) by \(27\) to achieve \(-54\).

By following these rules, we can find the correct sign for any multiplication problem.

Negative Numbers

Negative numbers often cause confusion, but they don’t have to! Let's break it down. Negative numbers are less than zero on the number line. When we say \(-x\), we mean the opposite of the variable \(x\).
In the expression \((-x)^3\), the meaning changes slightly based on \(x\)'s value.

• If \(x = -3\), \(-(-3)\) becomes \(3\).

It's important to note the difference between a negative sign outside an exponent and directly with the base, such as in \((-3)^3\) versus \(-3^3\).
The placement changes the calculation since negative bases raised to odd exponents result in negative outcomes.

Substitution

Substitution helps evaluate expressions by replacing variables with known values. It's a straightforward process: wherever you see the variable, replace it with its given number.
For the expression \(2(-1)(-x)^3\) with \(x = -3\), replace \(x\) by \(-3\).

• This transforms the expression into \(2(-1)(--3)^3\).
• Note that \(--3\) simplifies to \(3\) because of the double negation rule.

By systematically substituting and simplifying each part of the expression, we break down complex algebra problems into manageable steps that lead to a final solution.