Simplify by using the order of operations. Express your answer as a number.
$\frac{\left|(10-6)-3^3\right|}{69-24 \div 8 \cdot 5}=$
Answer (2)
The expression simplifies to 54 23 by evaluating the numerator and denominator step by step. The absolute value in the numerator yields 23, while the operations in the denominator result in 54. Thus, the final answer is 54 23 .
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Evaluate the expression inside the absolute value in the numerator: ( 10 − 6 ) − 3 3 = 4 − 27 = − 23 , then take the absolute value: ∣ − 23∣ = 23 .
Evaluate the division and multiplication in the denominator from left to right: 24 ÷ 8 ⋅ 5 = 3 ⋅ 5 = 15 , then subtract from 69: 69 − 15 = 54 .
Divide the simplified numerator by the simplified denominator: 54 23 .
The simplified expression is 54 23 .
Explanation
Understanding the Problem We are asked to simplify the expression 69 − 24 ÷ 8 ⋅ 5 ∣ ( 10 − 6 ) − 3 3 ∣ using the order of operations (PEMDAS/BODMAS). This means we need to evaluate the expression inside the parentheses and absolute value first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).
Simplifying the Numerator First, let's simplify the numerator. We start by evaluating the expression inside the absolute value. Inside the absolute value, we have ( 10 − 6 ) − 3 3 . We evaluate the parentheses first: 10 − 6 = 4 . Then we evaluate the exponent: 3 3 = 3 ⋅ 3 ⋅ 3 = 27 . Now we have ∣4 − 27∣ = ∣ − 23∣ . The absolute value of − 23 is 23 . So the numerator simplifies to 23 .
Simplifying the Denominator Next, let's simplify the denominator. We have 69 − 24 ÷ 8 ⋅ 5 . We perform division and multiplication from left to right. First, 24 ÷ 8 = 3 . Then, 3 ⋅ 5 = 15 . Now we have 69 − 15 = 54 . So the denominator simplifies to 54 .
Final Calculation Now we have the simplified expression 54 23 . This is already in simplest form, so we can write it as a decimal: 23 ÷ 54 ≈ 0.4259 .
Final Answer Therefore, the simplified expression is 54 23 .
Examples
Understanding the order of operations is crucial in many real-world scenarios, such as calculating financial transactions, designing engineering structures, or even planning a cooking recipe. For instance, if you're calculating the total cost of items with discounts and taxes, following the correct order ensures you get the accurate final amount. Similarly, in engineering, the order of operations is vital for precise calculations in structural designs to ensure safety and stability. In cooking, adding ingredients in the right sequence can determine the success of a dish.
