Problem 49 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ 2^{4}-2(2)^{3}-3(2)^{2}+7(2)-10 $$

Step-by-Step Solution

Answer

The simplified expression is -8.

Step 1: Evaluate the Exponents

First, let's resolve the exponents in the expression. \(2^4 = 16\),\(2^3 = 8\),\(2^2 = 4\).Substitute these back into the expression: \(16 - 2(8) - 3(4) + 7(2) - 10\).

Step 2: Simplify the Multiplications

Next, evaluate the multiplication operations: \(-2 \times 8 = -16\),\(-3 \times 4 = -12\),\(+7 \times 2 = 14\).Substitute the results into the expression: \(16 - 16 - 12 + 14 - 10\).

Step 3: Perform the Additions and Subtractions

Finally, perform the calculations in order:1. \(16 - 16 = 0\),2. \(0 - 12 = -12\),3. \(-12 + 14 = 2\),4. \(2 - 10 = -8\).

Step 4: Final Solution

The simplified expression evaluates to \(-8\).

Key Concepts

Simplifying Numerical Expressions

Simplifying numerical expressions combines arithmetic operations and algebra. That's adding, subtracting, multiplying, or dividing numbers until you can't simplify it further. In our example, we focus on the right steps to simplify correctly.

Start by carefully noting each number and operation. Ensure you keep track of signs (positive or negative) and follow the arithmetic order. Thoroughly check each calculation step by step. Simpler expressions are easier to manage. Following an established order is crucial for efficient simplifying.

Always aim to maintain clarity and correctness by following steps in sequence. This builds a strong foundation in algebra skills.

Evaluating Exponents

Exponents, also known as powers, show how many times a number is multiplied by itself. In the expression, there are exponents like \(2^4\), \(2^3\), and \(2^2\). Evaluating these is the first step.

• \(2^4 = 16\)
• \(2^3 = 8\)
• \(2^2 = 4\)

Substituting these values back into the expression replaces powers with numbers. It transforms it into basic arithmetic, which is easier to handle. Fermly understanding exponents boosts your capability in handling larger numbers and more complex algebraic expressions later on.

Remember, being precise in evaluating exponents affects the accuracy of further calculations.

Multiplication Operations

Multiplication follows evaluating exponents. Handle multiplication first when it appears alongside addition and subtraction. This rule is part of the order of operations, often remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• \(-2 \times 8 = -16\)
• \(-3 \times 4 = -12\)
• \(+7 \times 2 = 14\)

Recalculated values are placed back into the equation, simplifying it further. Always follow multiplication before moving to addition or subtraction.

Attention to detail during multiplication ensures each step leads accurately toward the correct final answer.

Addition and Subtraction in Algebra

After multiplication, move on to addition and subtraction. These operations tie everything together, leading to the final simplified result.

Start from the left, resolving each pair of numbers:

• \(16 - 16 = 0\)
• \(0 - 12 = -12\)
• \(-12 + 14 = 2\)
• \(2 - 10 = -8\)

Settling one step at a time avoids mistakes. Keep in mind the need for precision in these calculations. Missing details or misplacing numbers can change your answer entirely.

In the end, your proper handling of addition and subtraction delivers the correct result. This builds your reliability in problem-solving and strengthens algebraic skills.