Problem 36 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The expression simplifies to -28.

Step 1: Evaluate the Exponent

Start by evaluating the exponents in the expression. First, calculate \((-1)^2\). Since \((-1)\) is squared, which means multiplied by itself, you get: \((-1)\cdot(-1) = 1\).

Step 2: Simplify the First Term

Now, substitute the result from Step 1 back into the expression, replacing \((-1)^2\) with 1. The first term becomes \(-4(1)\), which simplifies to \(-4\cdot 1 = -4\).

Step 3: Evaluate the Second Exponent

Now calculate \(2^3\), which means \(2\) multiplied by itself three times: \(2 \cdot 2 \cdot 2 = 8\).

Step 4: Simplify the Second Term

Replace \(2^3\) with 8 in the expression. The second term is now \(-3(8)\), which simplifies to \(-3 \cdot 8 = -24\).

Step 5: Combine the Simplified Terms

Now combine both simplified terms from the expression: \(-4 - 24 = -28\).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in algebra that tell you how many times a number, known as the base, is multiplied by itself. For instance, in the term \((-1)^2\), the base is \(-1\) and the exponent is 2. Calculating an exponent is straightforward: multiply the base by itself as many times as the exponent specifies.

• \((-1)^2\) means \((-1) \, \text{multiplied by} \, (-1)\), resulting in 1.
• Similarly, in the expression \(2^3\), the base \(2\) is multiplied by itself three times: \(2 \cdot 2 \cdot 2 = 8\).

These calculations form the basis of simplifying expressions and help in performing complex algebraic manipulations.

The Art of Simplification

Simplification involves breaking down expressions into their simplest form while maintaining their value. This task involves various algebraic rules and strategies, often dealing with arithmetic operations like addition, subtraction, multiplication, and division.
When simplifying the expression \(-4(-1)^{2}-3(2)^{3}\), the first step is addressing the exponents as they dictate the multiplication involved. Once the exponents are processed, you focus on performing the remaining arithmetic operations:

• Replace each calculated power in the expression: \(-4(1)\) and \(-3(8)\).
• Continue simplifying: \(-4 \cdot 1 = -4\) and \(-3 \cdot 8 = -24\).

Finally, combine these results to obtain the simplest form of the expression, which is adding the computed values: \(-4 - 24 = -28\). Simplification enables clearer understanding and manipulation of algebraic expressions.

Decoding Algebraic Expressions

Algebraic expressions include numbers, variables, and arithmetic operations. They are the building blocks of algebra, allowing you to represent and solve equations. A deep understanding of algebraic expressions opens the door to solving complex mathematical challenges.
In the context of simplifying \(-4(-1)^{2}-3(2)^{3}\), recognizing each part of the expression is critical:

• The constants \(-4\) and \(-3\), which are coefficients in the terms.
• The numerical parts \((-1)^{2}\) and \(2^{3}\), which involve exponentiation.

By combining knowledge of arithmetic and algebraic rules, you accurately simplify the expression. Grasping how to evaluate these elements enables students to reduce the complexity of algebraic expressions and smooth out problem-solving processes in mathematics.