Problem 37 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$-3(-2)^{3}+4(-1)^{5}$$

Step-by-Step Solution

Answer

The simplified expression is 20.

Step 1: Calculate the Cubed Power

First, calculate the cube of \(-2\). This requires multiplying \(-2\) three times: \((-2)\times(-2)\times(-2) = -8\).

Step 2: Multiply with the Coefficient

Next, multiply the result from Step 1 by the coefficient \(-3\): \(-3 imes -8 = 24\).

Step 3: Calculate the Fifth Power

Now, calculate the fifth power of \(-1\). This requires multiplying \(-1\) five times: \((-1)\times(-1)\times(-1)\times(-1)\times(-1) = -1\).

Step 4: Multiply with the Coefficient

Take the result from Step 3 and multiply by the coefficient \(4\): \(4 imes -1 = -4\).

Step 5: Combine Results

Now, add the results from Step 2 and Step 4 together: \(24 + (-4) = 20\).

Key Concepts

Understanding Powers and Exponents

In mathematics, powers or exponents tell you how many times to multiply the base number by itself. For instance, the expression \((-2)^{3}\) means that \(-2\) is multiplied by itself three times: \((-2) \times (-2) \times (-2)\).

• When you have an even exponent, like \((-2)^2\), the result is positive because a negative number times a negative number is positive.
• Conversely, an odd exponent results in a negative product when multiplied together, as seen with \((-2)^{3} = -8\).

Exponents are vital in simplifying expressions, as they can dramatically change the value depending on the base and the power. Remember that both positive and negative numbers can have powers.

Handling Negative Numbers

Negative numbers can often be tricky, especially when combined with other negative numbers or exponents. In expressions like \(( -3) \times (-8)\), two negatives make a positive, resulting in \(24\). This principle arises because when a negative number is multiplied by another negative, the result flips back to positive.

• When multiplying or dividing two negative numbers, the result is positive.
• However, when multiplying a positive number by a negative, like \(4 \times (-1)\), the result is negative: \(-4\).

Always keep track of the signs to avoid mistakes, as incorrect handling of negatives leads to errors.

Applying the Order of Operations

The order of operations is crucial when simplifying numerical expressions, and it is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This helps in determining what calculations to perform first.
In the given exercise, you should first evaluate the powers: \((-2)^{3}\) and \((-1)^{5}\). After calculating the exponents, proceed with multiplying those results by their respective coefficients: \(-3\) and \(4\). Finally, add the results: \(24 + (-4)\).

• Remember to always deal with exponents before multiplication.
• Addition and subtraction should be the last operations performed after simplifying the rest of the expression.

This ensures the expression is simplified correctly and efficiently.