Problem 108 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{\left(-6^{2}-2^{4} \cdot 2\right)+5}{-4-3} $$

Step-by-Step Solution

Answer

The expression evaluates to 9.

Step 1: Evaluate Exponents

Begin by evaluating the exponents in the numerator of the expression. The expression involves two exponents: \[-6^2 = -(6 imes 6) = -36 \]and\[2^4 = 2 imes 2 imes 2 imes 2 = 16 \]

Step 2: Multiply the Terms

Next, multiply the result from the second exponentiation by 2 as per \[2^4 imes 2 = 16 imes 2 = 32 \]

Step 3: Calculate the Numerator

Now substitute the results from the previous steps into the numerator and evaluate: \[(-36 - 32) + 5 \]First, perform the subtraction inside the parentheses:\[-36 - 32 = -68 \]Then add 5:\[-68 + 5 = -63 \]

Step 4: Simplify the Denominator

Simplify the denominator of the expression, which is \[-4 - 3 = -7 \]

Step 5: Divide the Numerator by the Denominator

Divide the result from the numerator by the result from the denominator to evaluate the expression: \[\frac{-63}{-7} = 9 \]

Key Concepts

Order of Operations

When dealing with algebraic expressions, it's crucial to know the order of operations. This set of rules tells us the correct sequence to evaluate a mathematical expression. The standard order of operations can be remembered by the acronym PEMDAS:

• Parentheses - Perform any calculations inside parentheses first.
• Exponents - Solve powers or roots.
• Multiplication and Division (from left to right).
• Addition and Subtraction (from left to right).

In the provided exercise, the expression combines several operations including exponents, multiplicative terms, and subtraction and addition inside the fraction's numerator. Following PEMDAS ensures we solve the exponents first, proceed with multiplication, and finally compute addition or subtraction. Sticking to this order is key to arriving at the correct solution.

Exponents

Exponents in algebra signify that a number is multiplied by itself a certain number of times. For instance, in the expression presented, there are the terms \(-6^2\) and \(2^4\). The expression \-6^2\ means "negative six squared," not "negative of six squared," which is a common source of confusion. Let's explain further why these calculations result in \-36\ and \16\ respectively:

• For \(-6^2\), we calculate \(-(6 \times 6) = -36\).
• For \(2^4\), the base 2 is repeatedly multiplied: \(2 \times 2 \times 2 \times 2 = 16\).

Understanding how exponents function and correctly interpreting negative signs is crucial. Missteps here can lead to significant errors in solving algebraic expressions.

Numerator and Denominator

In fraction expressions, the terms 'numerator' and 'denominator' are fundamental. The numerator is the top part of a fraction that represents the number of parts of the whole, while the denominator is the bottom part, indicating into how many parts the whole is divided. In the expression \(\frac{(-63)}{-7}\), solving for the numerator involved evaluating \((-6^2 - 2^4 \cdot 2) + 5\).

Here's a breakdown:

• The result of the exponents and multiplication was \(-36 - 32 = -68\) and adding 5 got us \(-63\).
• The denominator was simplified directly by subtracting: \(-4 - 3 = -7\).
• Finally, the division \\(\frac{-63}{-7} = 9\) was calculated after addressing both parts individually.

Correctly simplifying numerators and denominators separately ensures that the final division gives you the correct result.