Problem 35 - Intermediate Algebra

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: Evaluate each exponential term

The first task is to evaluate the exponential expressions. Calculate \( (2)^3 \) and \( (-2)^3 \). \[(2)^3 = 2 \times 2 \times 2 = 8\]\[(-2)^3 = (-2) \times (-2) \times (-2) = -8\]

Step 2: Multiply each exponential result by its coefficient

Now multiply each evaluated exponential result by its respective coefficient: For \( 7(2)^3 \):\[7 \times 8 = 56\]For \( 4(-2)^3 \):\[4 \times (-8) = -32\]

Step 3: Add the results together

Finally, add the results from the previous calculations together to simplify the expression:\[56 + (-32) = 56 - 32 = 24\]

Key Concepts

Exponents

Exponents are a fundamental aspect of mathematical notation that allows you to represent repeated multiplication concisely and efficiently. When you see an expression like \((a)^n\), it essentially means "multiply \(a\) by itself \(n\) times." For example, \((2)^3\) means multiply \(2\) by itself three times: \(2 \times 2 \times 2 = 8\).

**Negative Bases:** When the base is negative, such as \((-2)^3\), it's important to pay attention to the number of times you multiply the base. An odd number of negations results in a negative product, so \((-2)\times (-2)\times (-2) = -8\).

**Key Points:**

• Exponents indicate how many times a number, called the base, is multiplied by itself.
• If the base is negative and the exponent is odd, the result is negative.
• These calculations are the first step when simplifying expressions that contain exponents.

Coefficients

Coefficients are numerical values placed in front of variables or terms that include exponents. They serve to scale the entire term. For example, in the expression \(7(2)^3\), the number '7' is the coefficient that multiplies the result of \((2)^3\).

**Understanding the Role:** Coefficients are critical in algebraic expressions because they determine how many times the term they accompany is counted. In our example, once you compute \((2)^3\), which equals 8, the coefficient instructs you to multiply that result by 7, which gives 56.

**Key Points:**

• Coefficients indicate the number of units of a variable or term.
• They can be positive or negative and affect the sign of the resulting product.
• Multiplying the evaluated exponent by its coefficient is essential before combining any terms.

Arithmetic Operations in Algebra

Arithmetic operations are the basic operations of addition, subtraction, multiplication, and division. In algebra, these operations are extended to include expressions and equations.

**Sequence of Operations:** Simplifying expressions follows a specific order known as the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This is crucial when dealing with complex expressions.

In the exercise in question, after evaluating the exponents and multiplying by their coefficients, you add or subtract these results. The order of operations ensures accuracy in simplification.

**Key Points:**

• Follow the correct order of operations to avoid mistakes.
• Combine like terms only after calculations involving exponents and coefficients are complete.
• Arithmetic operations give the final simplified form of an expression.