Problem 109 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ -\left(\frac{40-1^{3}-2^{4}}{3(2+5)+2}\right) $$

Step-by-Step Solution

Answer

The result is \(-1\).

Step 1: Simplify the Exponentiation Inside the Parentheses

First, calculate the powers inside the parentheses. We have two: - Calculate \(1^3\) which is \(1\) since any number raised to the power of 3 is itself.- Calculate \(2^4\) which is \(16\) since \(2 \times 2 \times 2 \times 2 = 16\).Thus, the expression inside becomes \(-\left(\frac{40 - 1 - 16}{3(2+5)+2}\right)\).

Step 2: Simplify the Subtraction Inside the Parentheses

Now, perform the subtractions in the numerator. - Subtract \(1\) and \(16\) from \(40\):\[40 - 1 - 16 = 23\].The expression now is \(-\left(\frac{23}{3(2+5)+2}\right)\).

Step 3: Simplify the Addition in the Denominator

Calculate the addition inside the parentheses in the denominator:- Add \(2\) and \(5\): \[2 + 5 = 7\].Now, the expression simplifies to \(-\left(\frac{23}{3 \times 7 + 2}\right)\).

Step 4: Evaluate the Multiplication in the Denominator

Multiply \(3\) by \(7\):- Calculate \(3 \times 7 = 21\).The expression is now \(-\left(\frac{23}{21 + 2}\right)\).

Step 5: Evaluate the Addition in the Denominator

Add \(21\) and \(2\):- \(21 + 2 = 23\).So, the expression simplifies to \(-\left(\frac{23}{23}\right)\).

Step 6: Divide the Numerator by the Denominator

Perform the division:- \(\frac{23}{23} = 1\).Replace the expression with \(-1\).

Step 7: Apply the Negative Sign

Finally, apply the negative sign from outside the parentheses:- The expression becomes \(-1\).

Key Concepts

Exponentiation

Exponentiation is one of the fundamental operations in mathematics. It involves raising a base number to a power or an exponent.
When you see a number written with a small number to its top right, that small number is called the exponent.
Here’s a simple overview:

• The base is the number you're multiplying.
• The exponent tells you how many times to multiply the base by itself.

For example, in the expression \(2^4\), 2 is the base, and 4 is the exponent.
This means you take 2 and multiply it by itself 3 more times (in total, four times):
\(2 \times 2 \times 2 \times 2 = 16\).
Remind yourself that multiplying a number by itself is called "raising to the power."
So, for \(1^3\), you're effectively calculating \(1 \times 1 \times 1\), which results in 1. Exponents simplify the expression and must be addressed first in calculations to ensure accurate results.

Order of Operations

The order of operations is a fundamental concept in mathematics. This set of rules tells us the correct sequence to evaluate a mathematical expression; without it, calculations can lead to different answers.
Here's a simple way to remember it: **PEMDAS**.

• **P**arentheses - Calculate anything in parentheses first.


• **E**xponents - Handle any exponents next.


• **M**ultiplication and **D**ivision - Go left to right; these operations are on the same level, so just tackle them in order.


• **A**ddition and **S**ubtraction - Like the previous step, do these last and order them from left to right.

In our exercise, this means after addressing any expressions within parentheses, we simplified the powers first, then followed multiplication and addition strictly by their operational hierarchy.

Parentheses Simplification

Parentheses play a crucial role in structuring expressions and can significantly alter the result of a calculation when placed or evaluated differently.
Simplifying parentheses involves evaluating any operations contained within first.

• This ensures whatever math is done inside can be treated as a single result.

Let's see the exercise: inside the function \(-\left(\frac{40-1^{3}-2^{4}}{3(2+5)+2}\right)\), everything within the outer parentheses must be taken care of first.
Therefore, we simplified what's within first: subtracting exponents under the numerator and breaking down the addition in the denominator.
So, through each step, you simplify layer by layer, allowing for the crux of your expression to be worked on without the clutter of complicated terms.