Problem 46 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$\left[-3(-1)^{3}-4(-2)^{2}\right]^{2}$$

Step-by-Step Solution

Answer

The expression simplifies to 169.

Step 1: Evaluate the Exponents

First, calculate the exponents inside the expression. For \((-1)^3\), we evaluate \((-1) imes (-1) imes (-1) = -1\). For \((-2)^2\), we evaluate \((-2) imes (-2) = 4\).

Step 2: Multiply Coefficients

Multiply the coefficients by the results of the exponents: \(-3(-1) = 3\) and \(-4 imes 4 = -16\).

Step 3: Simplify Inside the Brackets

Add the results from Step 2: \(3 - 16 = -13\). So the expression inside the brackets simplifies to \([-13]\).

Step 4: Square the Result

Square the result obtained in Step 3: \((-13)^2 = 169\).

Key Concepts

Exponentiation

Exponentiation is a fundamental operation in algebra that involves raising a number to the power of another number. This operation is expressed through the notation \(a^n\), where \(a\) is the base and \(n\) is the exponent. It indicates that the base \(a\) is multiplied by itself \(n\) times.

• An exponent of \(2\) indicates squaring the base, like \((-2)^2 = 4\).
• An exponent of \(3\) indicates cubing, as seen in \((-1)^3 = -1\).

Exponentiation follows a specific order in mathematical operations known as "PEMDAS" (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means exponents are calculated before any other operations, as evidenced by the need to evaluate \((-1)^3\) and \((-2)^2\) before going further in the simplification process. By performing exponentiation first, it ensures that you handle the smallest parts of the problem correctly, leading to an accurate overall result.

Simplification Process

The simplification process in algebra involves reducing an expression to its simplest form, thereby making it more manageable. This often requires a few steps to systematically break down the expression through different operations.To simplify complex expressions like our initial problem \([-3(-1)^{3}-4(-2)^{2}]^{2}\), you must follow these steps:

• Evaluate Exponents: As seen in the step-by-step solution, this is crucial since it affects the entire operation within the expression.
• Apply Multiplication: Multiply the results of the exponents by their coefficients, i.e., \(-3 \times (-1)\) and \(-4 \times 4\).
• Combine Like Terms: Add or subtract the results to simplify within any brackets or parentheses.
• Final Simplifications: In cases where you have brackets, carry out additional operations such as squaring the final result.

Each of these steps requires careful attention to detail, ensuring every calculation is accurately done before moving on to the next.

Numerical Expressions

Numerical expressions are combinations of numbers and operations (such as addition and multiplication) without variables. The goal is often to simplify these expressions to a single numerical value.The expression \([-3(-1)^{3}-4(-2)^{2}]^{2}\) holds various numerical components involving operations and exponents. Handling numerical expressions involves efficiently tackling these components as follows:

• Recognize the Structure: Understand parts of the expression, identifying exponents, and the sequence of operations to be performed.
• Evaluate Sequentially: As seen in the solution steps, each part of the expression must be resolved in a logical order—starting with exponents, followed by multiplication and addition/subtraction.
• Arrive at a Single Number: Once simplifications are applied correctly, the expression simplifies to one numerical value, such as \(169\) in our problem.

Through properly structured steps, handling numerical expressions becomes simpler and more intuitive, allowing a clear path from a complex expression to a concise answer.