Problem 97 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ -5(-2)^{3}-|-2+1| $$

Step-by-Step Solution

Answer

The expression evaluates to 39.

Step 1: Evaluate the Exponent

The expression is \(-5(-2)^{3} - |-2+1|\). First, calculate \((-2)^{3}\).Calculating\((-2)^{3} = (-2) imes (-2) imes (-2) = -8\).

Step 2: Multiply by -5

Now that we have \((-2)^{3} = -8\), we can substitute this value back into the expression:\(-5(-2)^{3} = -5(-8)\).Calculate this multiplication:\(-5 imes -8 = 40\).

Step 3: Simplify the Absolute Value

Next, simplify the absolute value \(|-2+1|\).Calculate \(-2+1 = -1\), and then take the absolute value:\(|-1| = 1\).

Step 4: Combine the Results

Now that we have calculated \(-5(-2)^{3} = 40\) and \(|-2+1| = 1\), substitute these values back into the expression:\(40 - 1\).

Step 5: Final Calculation

Subtract the simplified result of the absolute value from the result of the multiplication:\(40 - 1 = 39\).

Key Concepts

Exponentiation

Exponentiation may seem complex at first, but it's actually a straightforward concept. It involves multiplying a number by itself a certain number of times. For example, in the expression \((-2)^{3}\), the base is \(-2\) and the exponent is 3. This means you multiply \(-2\) by itself three times:

• First, multiply \(-2\) by \(-2\) to get 4.
• Next, multiply the result (4) by \(-2\) again to get -8.

So, \((-2)^{3} = -8\). The negative sign on the original base carries through because when you multiply an odd number of negative numbers, the result is negative. Keep this in mind: Odd exponents maintain the sign of a negative base, resulting in a negative number as they do here.

Absolute Value

The absolute value is a measure of distance from zero on the number line, regardless of direction. It converts any number into its non-negative version. Consider \(|-2+1|\). To solve it, follow these steps:

• First, handle the operation inside the absolute value: o - \(-2 + 1\) equals \(-1\).
• Now, apply the absolute value to \(-1\). Simply remove the negative sign, turning it into 1.

Thus, \(|-1| = 1\). Absolute value, denoted by two vertical bars, makes even negatives positive, since it focuses only on the size, not the sign.

Multiplication with Negative Numbers

Multiplying negative numbers can be tricky, but here are some tips to get it right easily. When you multiply numbers, follow straightforward rules:

• Two negative numbers multiplied result in a positive number. For example, \(-2 \times -2 = 4\).
• When you multiply a negative number by a positive number, the result remains negative, like \(-5 \times 2 = -10\).
• Similarly, a negative number times another negative results again in positive, as seen in \(-5 \times -8 = 40\).

Master these rules to handle negative numbers confidently. For expressions involving lots of computations, simply take each multiplication step by step, considering the sign of each factor. In the expression \(-5(-8)\), beginning with one positive and one negative indeed yields a positive 40. Keep these sign rules handy for such calculations.