Problem 48 - Elementary and Intermediate Algebra

Question

Simplify. $$ 2^{3}+2^{4}-5\left[8-4(9-10)^{2}\right] $$

Step-by-Step Solution

Answer

4

Step 1 - Simplify the Exponential Terms

Compute the exponentials: \[ 2^3 = 8 \] and \[ 2^4 = 16 \]. So the expression becomes: \[ 8 + 16 - 5\big[8 - 4(9 - 10)^2\big] \]

Step 2 - Simplify Inside the Parentheses

Compute the expression inside the innermost parentheses: \[ 9 - 10 = -1 \]. Next, square the result: \[ (-1)^2 = 1 \]. The expression now becomes: \[ 8 + 16 - 5\big[8 - 4 \times 1\big] \]

Step 3 - Multiply Inside the Brackets

Multiply 4 by the squared result: \[ 4 \times 1 = 4 \]. The expression becomes: \[ 8 + 16 - 5\big[8 - 4\big] \]

Step 4 - Simplify Inside the Brackets

Subtract inside the brackets: \[ 8 - 4 = 4 \]. The expression is now: \[ 8 + 16 - 5 \times 4 \]

Step 5 - Multiply

Multiply 5 and 4: \[ 5 \times 4 = 20 \]. The expression now is: \[ 8 + 16 - 20 \]

Step 6 - Add and Subtract

Finally, add and subtract in order: \[ 8 + 16 = 24 \], then \[ 24 - 20 = 4 \]

Key Concepts

Exponential Terms

Exponential terms involve numbers raised to a power, such as \(2^3\) and \(2^4\) in the given problem. Here, the base (2) is multiplied by itself as many times as the exponent dictates. For example, \(2^3\) means multiplying 2 by itself 3 times (2 x 2 x 2 = 8). Similarly, \(2^4\) means multiplying 2 by itself 4 times (2 x 2 x 2 x 2 = 16). These calculations are the first step in simplifying the expression.

By computing the exponential terms first, we reduce the complexity of the expression early on. So, our given expression simplifies from: \(2^3 + 2^4\) to 8 + 16, making the overall expression easier to handle. This emphasizes how handling exponentials first can simplify problem-solving in algebraic expressions.

Order of Operations

The order of operations is a set of rules to ensure consistent results when simplifying mathematical expressions. Remembering the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) can help:

• **P**arentheses first
• **E**xponents (like \(2^3\) and \(2^4\))
• **MD** (Multiplication and Division from left to right)
• **AS** (Addition and Subtraction from left to right)

In our problem, we have an expression \(2^3 + 2^4 - 5[8 - 4(9 - 10)^2]\). Right after handling the exponents, we evaluate the expression inside the parentheses first: solve \((9 - 10)\). Then square the result \((-1)^2 = 1\), followed by handling the multiplication in the brackets. Adhering to the correct order ensures the right outcome.

Skipping or rearranging these steps can lead to incorrect results. Always follow PEMDAS to make sure expressions simplify correctly.

Parentheses and Brackets

Parentheses and brackets are used to group parts of mathematical expressions that should be handled first. In our problem, parentheses appear inside the brackets. Start with the innermost parentheses, then work outward.

Here, evaluate \((9 - 10)\) first, resulting in -1. This value is then squared. By following the sequence (Parentheses and Brackets), each step becomes manageable, making the overall expression: \(8 + 16 - 5[8 - 4 \times 1]\). Next, handle the multiplication within the brackets to update to: \(8 + 16 - 5[4]\), finally simplifying it.

Using parentheses and brackets effectively clarifies the order in complex expressions, guiding each step toward the correct solution.