Problem 57 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ 2+6^{2} \div\left(3^{2}-5\right) $$

Step-by-Step Solution

Answer

11

Step 1 - Evaluate the exponent in the numerator

First, calculate the exponent in the expression. Here, we have to evaluate the exponent for 6 in the numerator: \(6^2 = 36\)

Step 2 - Evaluate the exponent in the denominator

Next, calculate the exponent in the denominator: \(3^2 = 9\)

Step 3 - Subtract in the denominator

Now, perform the subtraction in the denominator: \(9 - 5 = 4\)

Step 4 - Divide

Divide the result from the exponentiation in the numerator by the result from the subtraction in the denominator: \(36 \div 4 = 9\)

Step 5 - Add 2

Finally, add 2 to the result from the division: \(2 + 9 = 11\)

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number by itself. For instance, in the expression \(6^2\), the number 6 is the base, and 2 is the exponent.
To solve \(6^2\), you multiply 6 by itself: \(6 \times 6 = 36\). Similarly, for \(3^2\), multiply 3 by itself: \(3 \times 3 = 9\).

• Remember, perform all exponentiation before tackling other operations.
• This step ensures that we reduce complex expressions to simpler numbers.

Understanding exponents lays the groundwork for correctly following the order of operations in math problems.

Division

Division is one of the fundamental arithmetic operations. It involves splitting a number into equal parts. In our example, after solving the exponents, we get 36 and 4. We then perform the division step, \(36 \div 4\), which means distributing 36 into 4 equal parts, giving us 9.
Key points about division:

• Always follow the order of operations, dealing with parentheses and exponents first.
• Division often simplifies the problem, making it easier to combine with other operations later.
• Remember to handle dividends and divisors accurately to avoid mistakes.

Division helps in breaking down complex parts of mathematical problems into manageable results.

Subtraction

Subtraction is the process of taking one number away from another. It's essential in reducing expressions step by step. In the exercise, we subtract 5 from 9 (\(9 - 5\)), yielding 4.
Points to remember for subtraction:

• Ensure subtraction is done correctly, especially in multi-step problems.
• Subtraction is typically executed after parentheses and when simplifying within groups.
• Careful handling of subtractive steps is crucial for achieving accurate results.

Subtraction simplifies expressions, paving the way for subsequent operations.

Addition

Addition is combining two or more quantities to obtain a total. In our final step, we add 2 to 9 (\(2 + 9\)), resulting in 11. This step concludes our calculation.
Addition basics:

• Perform addition after completing other operations like parentheses, exponents, multiplication, and division unless parentheses dictate otherwise.
• Addition simplifies once the numbers involved are reduced through division or subtraction.
• It's the final step in many order of operations problems, bringing the solution to a close.

Addition finalizes the process, ensuring all parts of the problem have been correctly combined for the answer.