Problem 19 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ -6^{2}-3(2-4)^{4} $$

Step-by-Step Solution

Answer

The value is -84.

Step 1: Evaluate the Exponent

Start by evaluating the exponent within the term \(-6^{2}\), which means \(-(6^2)\). First calculate \(6^2\), which is \(36\). Since the negative sign is outside the exponent, it's \(-36\).

Step 2: Simplify the Parentheses

Simplify the part inside the parentheses of \(3(2-4)^{4}\). Calculate \(2-4\), which is \(-2\).

Step 3: Evaluate the Inner Exponent

Now that you have \(-2\) from the parentheses, you need to evaluate \((-2)^4\). \((-2)^4\) is \( (-2) imes (-2) imes (-2) imes (-2) = 16\).

Step 4: Apply the Coefficient

Multiply the result from Step 3 by the coefficient \(3\). This means calculating \(3 imes 16\), which equals \(48\).

Step 5: Compute the Final Result

Combine the results from Steps 1 and 4. You have \(-36\) from Step 1 and \(48\) from Step 4, which gives you \(-36 - 48 = -84\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving the raising of a number (the base) to a power (the exponent). It is a shortcut for repeated multiplication. For example, when you see \(6^2\), it means 6 multiplied by itself: \(6 \times 6 = 36\). This part of an expression is crucial to evaluate correctly.

In the given exercise, we handle the term \(-6^2\). It's important to note that the negative sign here is outside the exponent. Thus, you only apply the power to the 6 and not to the 6 with the minus. Therefore, it becomes \(-(6^2)\) or \(-36\). This is as if you multiply \(-1\) and the result of the exponentiation \(36\).

• Always ensure whether the negative is inside or outside the exponent.
• Calculate the base raised to the exponent first.

Order of Operations

The order of operations is a collection of rules used to clarify which procedures should be performed first in a given mathematical expression. A common mnemonic for remembering the order is PEMDAS:

• P: Parentheses
• E: Exponents
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

In practical terms, this means you should always begin by solving expressions inside parentheses, then calculate exponents, followed by multiplication and division, and finally, perform addition and subtraction.

For our exercise, you first manage the parentheses: \(2-4\), which simplifies to \(-2\). Then solve any exponents: \((-2)^4\). Only after these steps should you conduct multiplication with the coefficient and subtraction. This order ensures that the operation respects mathematical conventions and yields the correct result.

Always double-check your steps closely to avoid simple mistakes and ensure accuracy.

Negative Numbers

Negative numbers often represent quantities that are less than zero. They are essential in scenarios like temperature below freezing or debt. Handling negative numbers requires an understanding of basic rules:

• When multiplying or dividing, a negative times a positive yields a negative, but a negative times a negative gives a positive result.
• When adding or subtracting, think of a number line: moving left means becoming more negative, right means more positive.

In the expression \(-6^2 - 3(2-4)^4\), the number \(-6^2\) involves evaluating the squaring of 6, resulting in 36 with an external negative changing it to \(-36\). Similarly, inside the parentheses, \(2-4\) yields \(-2\), which when raised to the power 4, results in a positive value (16) because multiplying negatives four times (an even number) gives a positive result.

Finally, deal with negatives carefully when adding \(-36 - 48\), minding to subtract more, landing at \(-84\). This clear handling ensures you don't misplace your negative signs and helps in maintaining accuracy.