Problem 88 - Beginning and Intermediate Algebra

Question

Calculate. $$5\left[3-8 \cdot 3^{2}+4 \cdot 6-2\right]$$

Step-by-Step Solution

Answer

The simplified expression is \(-235\).

Step 1: Evaluate the exponent inside the brackets

Evaluate the exponent \(3^2\): \(3^2 = 3 \cdot 3 = 9\) So the expression becomes: \(5[3-8 \cdot 9 + 4\cdot 6 - 2]\)

Step 2: Compute Multiplication inside the brackets

Perform both multiplications inside the brackets: - \(8\cdot 9 = 72\) - \(4\cdot 6 = 24\) The expression now becomes: \(5[3-72 + 24 - 2]\)

Step 3: Perform Addition and Subtraction inside the brackets

Compute the sum/difference of the numbers inside the brackets: \(3 - 72 + 24 - 2 = (-69) + 24 - 2 = -45 - 2 = -47\) Now the expression becomes: \(5[-47]\)

Step 4: Multiply the result by 5

Finally, multiply the expression by 5: \(5[-47] = -47 \cdot 5 = -235\) So, the simplified expression is: \(-235\).

Key Concepts

Exponents

Exponents are an essential mathematical concept that involves raising a number to a certain power. In our exercise, we encounter the exponent \(3^2\), which is read as "three squared." This means that 3 is multiplied by itself.

• The base number 3 is the number being multiplied.
• The exponent 2 indicates how many times the base is used as a factor.

So, \(3^2\) is calculated as \(3 \cdot 3 = 9\). This operation reduces the complexity within the brackets and allows us to proceed with multiplication steps. Handling exponents first is essential for simplifying expressions correctly, as they often alter the other operations.

Multiplication

Once exponents have been simplified, the next step in the order of operations is multiplication. In our expression within the brackets, we tackle two multiplication problems: \(8 \cdot 9\) and \(4 \cdot 6\).
Multiplication is a straightforward process:

• For \(8 \cdot 9\), multiply 8 and 9 to get 72.
• For \(4 \cdot 6\), multiply 4 and 6 to get 24.

These products replace the original multiplication expressions, simplifying our entire calculation. Multiplication needs to be done before addition and subtraction to ensure the expression's correct evaluation. This step prepares the expression for the addition and subtraction process.

Addition and Subtraction

After handling multiplication, we move on to addition and subtraction within the brackets. According to the order of operations, these are performed from left to right. In our expression, the operations involve solving: \(3 - 72 + 24 - 2\).

• First, subtract 72 from 3 to get -69.
• Next, add 24 to -69, resulting in -45.
• Finally, subtract 2 from -45 to get -47.

Handling these operations in sequence ensures accuracy.
It's crucial not to mix up the order or miss steps when dealing with multiple operations, as each change affects the resulting value. Adding and subtracting last helps clarify the expression's numeric relationships.

Brackets

Brackets are used in mathematics to specify the operations that need to be executed first. They take precedence over other operations. Our entire calculation is initially placed inside brackets: \(5[3-8 \cdot 3^{2}+4 \cdot 6-2]\).
The steps inside the brackets must be completed before considering any operations outside, such as the multiplication by 5.

• First, solve exponents and multiplications within brackets.
• Then, proceed with addition and subtraction.

By following these steps, we ensure that all internal operations are entirely completed. Finally, the resulting number from these calculations, -47 in this case, is then multiplied by 5 outside the brackets, yielding the final answer of -235. Brackets ensure accuracy by clearly defining the order and the scope of operations.