Problem 35 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$7(2)^{3}+4(-2)^{3}$$

Step-by-Step Solution

Answer

The simplified expression is 24.

Step 1: Calculate the powers

First, we need to calculate the powers in the expression. Calculate \( (2)^3 \) and \( (-2)^3 \). \((2)^3 = 2 \times 2 \times 2 = 8\) and \((-2)^3 = (-2) \times (-2) \times (-2) = -8\).

Step 2: Substitute the powers into the expression

Now substitute the values we found back into the original expression. This gives us: \(7(8) + 4(-8)\).

Step 3: Perform the multiplication

Now perform the multiplication for each term. \(7(8) = 56\) and \(4(-8) = -32\). Replace these into the expression: \(56 + (-32)\).

Step 4: Simplify by adding the terms

Finally, add the results from Step 3 together: \(56 + (-32) = 24\).

Key Concepts

Exponents

When dealing with exponential expressions, you'll encounter numbers raised to a power. An exponent indicates how many times to multiply the base by itself. For example, in the expression \((2)^3\), the number 2 is the base, and the exponent is 3. This means you multiply 2 by itself three times, which is expressed mathematically as \(2 \times 2 \times 2 = 8\).

It's important to recognize the impact of a negative base in exponents. For \((-2)^3\), the same rule applies: Multiply the base, -2, by itself three times. Hence, \((-2) \times (-2) \times (-2) = -8\). The odd exponent retains the base's negativity, resulting in a negative product. Understanding how to manipulate and calculate exponents is crucial in simplifying expressions.

Mastering exponents helps in performing more complex algebraic operations, as it lays the foundation for understanding growth rates, scientific notation, and various functions in advanced mathematics.

Multiplication in Algebra

Multiplication in algebra often includes variables and constants, as well as operations with constants, like in our initial problem. Here, once we calculate the powers of the terms, we use multiplication to combine them with any coefficients present before moving on to other operations.

Consider the expression \(7(2)^3 + 4(-2)^3\) after calculating the powers: we substitute \(8\) and \(-8\) from earlier computations. This transforms the expression to \(7(8) + 4(-8)\). Here multiplication is straightforward.

• Multiply the coefficient by the power: \(7 \times 8 = 56\) and \(4 \times (-8) = -32\).

The outcome of these multiplicative operations is inserted back into the expression, facilitating the simplification process.

Understanding these steps is vital as algebraic multiplication underpins many different types of problems, such as those involving polynomials and more complex mathematical structures.

Integer Operations

Working with integers requires careful attention to adding and subtracting results, especially when involving negative and positive numbers. In the original problem, after performing multiplication, we're left with \(56 + (-32)\). When adding integers with different signs, it's essential to follow these rules:

• If one number is negative and the other positive, essentially subtract the smaller numerical value from the larger one and keep the sign of the larger number's value.
• So, \(56 + (-32)\) simplifies to \(56 - 32\), requiring you to subtract the numbers and retain the positive sign, resulting in \(24\).

Handling operations with integers, especially with negative numbers, is a fundamental skill in math. Not only does it help you simplify expressions like the one discussed, but it also assists in resolving real-world problems, like calculating temperatures, financial transactions, and beyond.