Problem 88 - Elementary Algebra

Question

Find the value of \(\frac{-3(4-9)-6(-3)-1}{2^{3}}\).

Step-by-Step Solution

Answer

Answer: 4

Step 1: Work Inside the Parentheses

Calculate the value inside parentheses first: \((4-9)=-5\)

Step 2: Apply Multiplication and Division

Apply the multiplication and division in the following order: 1. Multiply -3 by the value inside the parentheses: \(-3(-5) = 15\) 2. Multiply -6 by -3: \(-6(-3) = 18\) 3. Find the value of \(2^{3}\): \(2^{3}= 2*2*2 = 8\)

Step 3: Add and Subtract Values

Add and subtract the values from Step 2: 1. Add the result of the first multiplication to the result of the second multiplication: \(15 + 18 = 33\) 2. Subtract 1 from the sum: \(33 - 1 = 32\)

Step 4: Perform the Division

Divide 32 by the result of \(2^{3}\) (which is 8): \(\frac{32}{8} = 4\) The value of the given expression \(\frac{-3(4-9)-6(-3)-1}{2^{3}}\) is equal to 4.

Key Concepts

Order of Operations

When approaching any algebraic expression, following the order of operations is essential. The sequence is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This acronym helps ensure that calculations are performed in the correct sequence:

• Start by solving whatever is inside any parentheses first.
• Next, handle any exponents (such as squaring a number or raising to a power).
• Move on to multiplication and division, proceeding from left to right in the expression.
• Finally, tackle addition and subtraction, also from left to right.

By following this structure, you can avoid common mistakes and arrive at the correct answer. In the given exercise, the calculation adhered to this order by first addressing the contents within the parentheses, followed by applying multiplication and division, and ending with addition and subtraction.

Integer Arithmetic

Understanding integer arithmetic is crucial when performing calculations involving whole numbers, both positive and negative. Integer arithmetic follows specific rules that must be respected to ensure accuracy in calculations. Here's a quick overview:

• Adding two positive numbers always yields a positive sum.
• Adding two negative numbers results in a negative sum.
• Adding a positive and a negative number involves subtraction, and the sign of the result will be the same as the number with the larger absolute value.
• Subtracting is the same as adding the opposite: switch the subtracted number's sign and then add.
• Multiplying two numbers of the same sign gives a positive product.
• Multiplying two numbers of different signs gives a negative product.

In the exercise, apply these rules to multiplication operations correctly. Multiplying \(-3(-5)\) resulted in a positive 15, and similarly, \(-6(-3)\) also yielded a positive 18.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to a power. The power, or exponent, determines how many times the base is multiplied by itself. For instance, the expression \(2^3\) means multiplying 2 by itself three times:

• \(2 \times 2 \times 2 = 8\).

It's important to pay careful attention to the base and the exponent, especially when the base is negative. The rules of exponentiation mean that:

• A positive base with a positive integer exponent results in a positive value.
• A negative base raised to an even exponent is positive, while if raised to an odd exponent, it remains negative.

In the exercise, \(2^3\) was calculated as 8, which later served as the divisor in the final step. Always perform exponentiations early, as dictated by the order of operations guidelines.