Problem 59 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \left(2^{4}+2^{5}-2^{3} \cdot 5\right)^{2} \div 4^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression using the order of operations: $$ \left(2^{4}+2^{5}-2^{3} \cdot 5\right)^{2} \div 4^{2} $$ Answer: 4

Step 1: Evaluate the exponents

First, we need to evaluate the powers of 2 and 4. Using the given expression: $$ \left(2^{4}+2^{5}-2^{3} \cdot 5\right)^{2} \div 4^{2} $$ Calculate the exponents: $$ \left(16+32-8 \cdot 5\right)^{2} \div 16 $$

Step 2: Perform addition, subtraction, and multiplication operations

Now, we need to perform the operations inside the parentheses. Follow the order of operations, which states that we must perform multiplication before addition and subtraction: $$ \left(16+32-40\right)^{2} \div 16 $$ Now, perform the addition and subtraction operations: $$ \left(8\right)^{2} \div 16 $$

Step 3: Square the result

Next, square the result inside the parentheses: $$ 64 \div 16 $$

Step 4: Divide

Finally, divide the two numbers to simplify the expression: $$ \frac{64}{16} = 4 $$ The simplified expression is 4.

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that represent repeated multiplication of a base number. In the expression \(2^4\), the number 2 is the base, and the 4 is the exponent, signifying that 2 should be multiplied by itself four times, yielding 16. Similarly, matters like \(2^5\) mean you multiply 2 by itself five times to reach 32. When evaluating expressions with exponents, it is important to follow the order of operations and calculate powers at the necessary stage. This order is crucial in ensuring accurate and consistent results when simplifying mathematical expressions. In the given problem, solving the exponents first transforms the equation into a more manageable form to apply further mathematical operations.

Multiplication and Division

After evaluating the exponents, multiplication and division come next in the order of operations. These operations must be approached from left to right as they appear in the expression. In the presented problem, multiplication happens with \(2^3 \cdot 5\), which simplifies to 40 after calculating \(2^3 = 8\) and multiplying it by 5. After completing all operations inside the parentheses, division follows after squaring the inner expression. In the final step of simplifying, \(64 \div 16\) yields the value of 4. Following the accurate sequence of multiplication and division ensures correctness in the entire problem-solving process.

Addition and Subtraction

Addition and subtraction are the last steps in the PEMDAS/BODMAS order of operations. After handling exponents and any multiplication or division, addition and subtraction are executed from left to right. In our problem, addition and subtraction occur within the parentheses after multiplying 8 by 5. Inside the parentheses, we add 16 to 32 and subtract 40: \(16 + 32 - 40\). This simplifies to 8, perfect for subsequent calculations. It's crucial to dampen the natural inclination to immediately tackle these operations without respecting the hierarchy set by PEMDAS/BODMAS. By following the right sequence, one gets accurate results consistently; here, moving step-by-step through the operations led us to our final answer of 4, post the concluding division.