Problem 44 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(2^{2} \cdot 3+2^{3} \cdot(6-2)-(3+17)+11(6)\)

Step-by-Step Solution

Answer

The final value is 90.

Step 1: Simplify the Exponents

Start by evaluating the exponential expressions. Compute each power separately.\[2^2 = 4\]\[2^3 = 8\]

Step 2: Resolve Expressions Inside Parentheses

Next, evaluate the expression inside the parentheses:\[6 - 2 = 4\]

Step 3: Substitute and Calculate Multiplications

Substitute the evaluated values back into the expression and perform the multiplications:\[4 \cdot 3 = 12\]\[8 \cdot 4 = 32\]\[11 \cdot 6 = 66\]

Step 4: Combine Terms

Substitute all the calculated values back into the expression and perform the remaining additions and subtractions:\[12 + 32 - 3 - 17 + 66\]

Step 5: Calculate the Final Result

Add and subtract the remaining terms to obtain the final result:\[12 + 32 = 44\]\[44 - 3 = 41\]\[41 - 17 = 24\]\[24 + 66 = 90\]

Step 6: Verify Using a Calculator

To ensure accuracy, check your result with a calculator. Input the entire expression and verify that the result matches:\[2^{2} \cdot 3 + 2^{3} \cdot(6-2)-(3+17)+11(6) = 90\]

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number. In the exercise, we have two exponents, namely \(2^2\) and \(2^3\). The expression \(2^2\) means that you multiply 2 by itself:

• \(2 \times 2 = 4\)

Similarly, \(2^3\) means that you multiply 2 three times:

• \(2 \times 2 \times 2 = 8\)

Exponents are always solved first in an equation according to the order of operations. This means before simplifying the expression inside parentheses or doing any multiplication, evaluate any exponents that are present.
Using exponents correctly is important because it allows for expressing large numbers in a compact form and simplifies complex expressions efficiently.

Parentheses

Parentheses are used in mathematical expressions to indicate that the operations within them should be performed first before any outside operations. In the given exercise, we have the expression \((6-2)\). The parentheses around \(6 - 2\) tell us to calculate this subtraction first:

• \(6 - 2 = 4\)

After simplifying within the parentheses, the result \(4\) is then used in later calculations.
Parentheses can greatly change the outcome of an expression by affecting the order in which operations are performed. Other than altering priorities in calculations, they also help in clearly organizing expressions, making them easier to read and understand.

Multiplication

Multiplication is one of the fundamental operations in mathematics, used to quickly add a number to itself a specified number of times. From the original problem, after solving the exponents and parentheses, we substitute the results back into the expression.
The expression now has three multiplication terms:

• \(4 \cdot 3 = 12\)
• \(8 \cdot 4 = 32\)
• \(11 \cdot 6 = 66\)

Perform each multiplication over other operations, except expressions within parentheses or exponents. This aligns with the rules of the order of operations where multiplication occurs after parentheses and exponents, but before addition and subtraction.

Addition and Subtraction

Addition and subtraction are operations that combine or remove quantities and are performed last in the order of operations, after parentheses, exponents, and multiplication/division. In the exercise, after calculating the multiplications, the remaining task is to simplify the final terms.
We perform the following steps:

• Add the results of the first two operations: \(12 + 32 = 44\)
• Subtract \(3\): \(44 - 3 = 41\)
• Subtract \(17\): \(41 - 17 = 24\)
• Lastly, add \(66\): \(24 + 66 = 90\)

By following the correct order, which is essential for solving expressions accurately, we reach the correct final answer. The overall solution is verified through careful calculation and checking with a calculator.