Problem 95 - Elementary Algebra

Question

\(5^{3}+3 \cdot 8 \div(-4-2)\)

Step-by-Step Solution

Answer

121

Step 1: Simplify the Exponent

First, calculate the value of the exponent, which is the term with the power. In this case, it is the term with the base 5 and exponent 3: \( 5^{3} = 5 \cdot 5 \cdot 5 = 125 \)

Step 2: Evaluate the Parentheses

Next, evaluate the expression inside the parentheses, which is \( -4 - 2 = -6 \)

Step 3: Perform the Division

Now, calculate the division part of the expression: \( 3 \cdot 8 \div (-6) \). First, perform the multiplication: \( 3 \cdot 8 = 24 \). Then, perform the division: \( 24 \div (-6) = -4 \)

Step 4: Final Addition

Finally, sum the results from Steps 1 and 3: \( 125 + (-4) = 121 \)

Key Concepts

Exponents

Exponents are a fundamental part of mathematics that indicate how many times a number, known as the base, is multiplied by itself. In our problem, the base is 5, and the exponent is 3. We write this as \(5^3\). This tells us to multiply 5 by itself three times: \(5 \cdot 5 \cdot 5\). When we calculate this, we get 125. It's important to handle exponents first in any mathematical expression to follow the proper order of operations.

Parentheses

Parentheses contain expressions that need to be simplified before anything outside is tackled. In our exercise, we have the expression \(-4 - 2\) inside parentheses. According to the order of operations, we simplify this first to \(-6\). Parentheses are essentially a way to group terms that must be addressed before any other operations outside the parentheses can be executed. Always deal with the contents inside parentheses first to avoid mistakes in the calculation.

Multiplication and Division

After handling exponents and parentheses, we move to multiplication and division, which are performed from left to right. In the problem, we have \(3 \cdot 8 \div (-6)\). First, we multiply 3 by 8, which gives us 24. Then, we divide 24 by -6, resulting in -4. Remember, the order in which you perform these operations—left to right—is crucial. Multiplication and division are performed on an equal level of priority.

Addition and Subtraction

Finally, we address addition and subtraction, the last operations in the order of operations. From our steps, we are left with adding 125 and -4. When you add a positive number to a negative number, it's similar to subtracting: \(125 + (-4) = 121\). Always handle any addition and subtraction after all other operations are completed. Understanding this ensures you get the correct outcome in multi-step problems.