Problem 104 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ (-2)^{3}\left(\frac{-6}{2}\right)(-1) $$

Step-by-Step Solution

Answer

The expression evaluates to -24.

Step 1: Evaluate the Exponent

Start by evaluating the exponent in the expression. \[ (-2)^3 = (-2) \times (-2) \times (-2) = -8 \]

Step 2: Simplify the Fraction

Next, simplify the fraction in the expression. \[ \frac{-6}{2} = -3 \]

Step 3: Multiply the Results

Now, multiply the results from Step 1 and Step 2, and then multiply the third term.First, multiply the results:\[ -8 \times (-3) = 24 \]Then multiply the result by the last term:\[ 24 \times (-1) = -24 \]

Step 4: Conclusion

The expression \((-2)^{3}\left(\frac{-6}{2}\right)(-1)\) evaluates to: \[-24\]

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, in the expression \((-2)^3\), the exponent is 3, which tells us to multiply \(-2\) by itself three times:

• \(-2 \times -2 = 4\)
• Then, \(4 \times -2 = -8\)

This means that \((-2)^3 = -8\). The negative sign stays because the base is negative and it's raised to an odd exponent. An important thing to remember is:

• An even exponent with a negative base results in a positive number.
• An odd exponent with a negative base results in a negative number.

By understanding these basic rules, you can simplify expressions with exponents effectively.

Fraction Simplification Basics

Fraction simplification is the process of reducing a fraction to its simplest form. This means making the numerator and the denominator as small as possible while still having the same value. In the original expression, we find the fraction \(\frac{-6}{2}\).
To simplify:

• Divide both the numerator (-6) and the denominator (2) by their greatest common divisor (GCD).
• The GCD of 6 and 2 is 2.
• So, divide: \(-6 \div 2 = -3\) and \(2 \div 2 = 1\).

This means that the simplified fraction is \(-3\) because the denominator of 1 can be omitted in multiplication or standalone expressions. Remember, simplifying fractions helps in making calculations much easier and prevents handling unnecessarily large numbers.

Mastering the Order of Operations

The order of operations is a set of rules followed to ensure mathematical expressions are solved correctly. This concept is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When solving an expression like \((-2)^{3}\left(\frac{-6}{2}\right)(-1)\), begin by handling the parts of the expression in order:

• First, solve the exponent, \((-2)^3 = -8\).
• Next, simplify and resolve the fraction, \(\frac{-6}{2} = -3\).
• Then, perform multiplication from left to right: \(-8 \times -3 = 24\).
• Finally, multiply that result by the last part: \(24 \times -1 = -24\).

By consistently following these rules, one can accurately evaluate complex expressions. Knowing the order of operations ensures you solve mathematical problems accurately and efficiently.