Problem 98 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ 6(-3)^{3}-|-6+5| $$

Step-by-Step Solution

Answer

The expression evaluates to -163.

Step 1: Evaluate the Power

Start by evaluating the power in the expression. The base is \(-3\) and the exponent is \(3\). Calculate \((-3)^3\): \[(-3) \times (-3) \times (-3) = -27\]So, \((-3)^3 = -27\).

Step 2: Multiply by 6

Now, multiply the result of the power by 6 as per the expression: \[6 \times (-27) = -162\]

Step 3: Simplify Absolute Value

Next, evaluate the absolute value expression \(-6 + 5\): \[-6 + 5 = -1\]The absolute value of \(-1\) is \| -1 \| = 1.

Step 4: Final Computation

Combine the results from Step 2 and Step 3 in the expression, completing the computation: \[-162 - 1 = -163\] Therefore, the expression evaluates to \(-163\).

Key Concepts

Understanding Exponents

Exponents are an essential part of mathematics, indicating how many times to multiply a number by itself. In our expression, we encountered

• Base:

  Here, the base is \( -3 \), the number that will be multiplied repeatedly.

• Exponent:

  The exponent is \( 3 \), which tells us to multiply the base by itself three times.

To solve \((-3)^3\), multiply \(-3 \times -3 \times -3\). The first two \((-3)\) give you \(9\), since a negative times a negative is positive. Then multiply \(9\) by \(-3\) again, resulting in \(-27\). Therefore, \((-3)^3\) equals \(-27\). This demonstrates how the sign of the base, when raised to an odd exponent, results in a negative product. Remember, always handle exponents before moving on to other operations, as dictated by the order of operations, often abbreviated as PEMDAS.

Absolute Value Context

Absolute value relates to the distance of a number from zero on the number line, always rendered as a non-negative number, regardless of a positive or negative input. Consider the expression \(-6 + 5\).

First, solve inside the absolute value bars: \(-6 + 5 = -1\).

The absolute value of \(-1\) is represented as \(|-1|\).

This calculates to \(1\), as absolute value strips away any negative sign.

Understanding absolute values is crucial when evaluating expressions that involve both positive and negative numbers, as it simplifies them to their magnitudes. Knowing how to quickly change any result inside the absolute value to its non-negative form helps in maintaining proper order in computations.

Integer Multiplication Basics

Multiplication involving integers can introduce multiple concepts, especially when negative numbers are involved. In our expression, after calculating the power, we multiplied \(-27\) by \(6\).

• Multiplication of two numbers:

  Calculate normally, ignoring the signs initially: \(27 \times 6\) is \(162\).

• Sign rules:

  Since \(-27\) is negative and \(6\) is positive, multiplying them results in \(-162\). The product of a positive and a negative is always negative.

Always remember, the product will only be positive if the integers have the same sign (both positive or both negative). More importantly, keep negative and positive integers distinct to maintain accuracy in your final results, especially when trending through complex calculations involving multiple steps like our example.