Problem 63 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ -2\left[2+4^{2}(8-9)\right]^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to -392.

Step 1: Simplify inside the innermost parentheses

First, evaluate the expression inside the parentheses: \(8 - 9 = -1\). So, replace \(8-9\) with \(-1\).

Step 2: Evaluate the exponent

Now substitute the simplified value into the expression and evaluate the exponent: \(4^{2}(-1)\). Calculate \(4^{2} = 16\).

Step 3: Multiply by the result from Step 1

Since \(4^{2}(-1)\) is next, multiply:\(16 imes (-1) = -16\). Replace the existing expression with \(-2[2 + (-16)]^{2}\).

Step 4: Simplify the expression inside the brackets

Now, further simplify inside the brackets:\(2 + (-16) = -14\). The expression now reads: \(-2(-14)^{2}\).

Step 5: Calculate the square of the number inside the brackets

Next, calculate \((-14)^{2}\):\((-14) imes (-14) = 196\). This transforms the overall expression to \(-2 imes 196\).

Step 6: Final multiplication to evaluate the expression

Finally, multiply the coefficients:\(-2 imes 196 = -392\).

Key Concepts

Order of Operations

When evaluating algebraic expressions, it's crucial to follow the correct sequence known as the "Order of Operations." This order ensures that everyone solves the expression in the same way, arriving at the correct answer.
The Order of Operations is often remembered by the acronym PEMDAS, which stands for:

• Parentheses - Solve expressions inside parentheses first.
• Exponents - Calculate all exponential terms next.
• Multiplication and Division - From left to right, perform these operations next.
• Addition and Subtraction - Finally, perform these actions from left to right.

To evaluate \(-2[2 + 4^{2}(8-9)]^{2}\), we start with the innermost parentheses, \((8-9)\), as instructed by the "P" in PEMDAS. Next, we address the exponent and follow through the list.
Sticking to this order avoids errors, especially in complex expressions.

Exponentiation in Algebra

Exponents in algebra are a way to express repeated multiplication. When you see an expression like \(4^{2}\), it means 4 multiplied by itself: \(4 \times 4 = 16\). Understanding how to handle exponents is vital when evaluating algebraic expressions.
In our original exercise, evaluating \(4^{2}\) was one of the key steps. After calculating the exponent, you know the base value during subsequent operations.
Remember, exponents are indicated in the order of operations after parentheses but before multiplication, division, addition, and subtraction. This position emphasizes their priority, making it crucial to calculate them right after dealing with any present parentheses.

Negative Numbers in Algebra Expressions

Handling negative numbers in algebra requires careful attention as mistakes with signs can lead you to incorrect answers. Negative numbers come into play frequently and can alter the result of calculations dramatically.
When managing expressions like \(-1\) or multiplying by negative numbers like in our exercise, remember:

• A negative number multiplied by a positive number results in a negative number.
• A negative number multiplied by another negative number results in a positive number.
• Addition and subtraction rules obey the number line direction, meaning expanding or reducing values accordingly.

In the given problem, after subtracting inside the parentheses, which resulted in \(-1\), careful handling of further steps with negatives was essential. Also, during the multiplication of \(-2 \times 196\), understanding how negative interacts with positive number factors became critical.
Properly managing negative numbers ensures accuracy in expressions and avoids mistakes.