Problem 29 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. $$15^{2}+5^{2} \cdot 2^{2}$$

Step-by-Step Solution

Answer

325

Step 1: Identify the Order of Operations

Recall the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This equation contains exponents, multiplication, and addition.

Step 2: Solve the Exponents

Identify and solve all exponent operations first: \[15^{2} = 225 \]\[5^{2} = 25 \]\[2^{2} = 4 \]

Step 3: Perform Multiplication

Using the results from Step 2, proceed to multiplication within the equation: \[25 \cdot 4 = 100 \]

Step 4: Perform Addition

Now, add the result from Step 3 to the value obtained from \(15^{2}\):\[225 + 100 = 325 \]

Step 5: Conclusion

The final value of the expression \(15^{2} + 5^{2} \cdot 2^{2}\) is \(325\).

Key Concepts

Understanding PEMDAS

The order of operations is a critical concept in mathematics that dictates how to solve expressions properly. When faced with a complex mathematical expression, performing operations in the incorrect order can lead to wrong results. This is where PEMDAS comes in, providing a structured guide for tackling these problems. The acronym PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

It's like a roadmap that ensures every mathematician gets to the correct destination. By following PEMDAS, you can systematically simplify and solve any mathematical equations. In our exercise, we'll see how PEMDAS helps us solve an expression with exponents, multiplication, and addition.

Solving with Exponents

Exponents represent repeated multiplication of a number by itself. For instance, in the expression given, we have the numbers 15, 5, and 2 each raised to the power of 2, known as squaring the numbers. This means:

• You multiply 15 by itself: \[15^{2} = 15 \times 15 = 225\]
• For 5:\[5^{2} = 5 \times 5 = 25\]
• And for 2:\[2^{2} = 2 \times 2 = 4\]

These computations simplify our original problem significantly. Exponents indicate a powerful operation, allowing us to replace a complex sequence of multiplications with a single number.

Applying Multiplication and Addition

Once you have simplified the expression using exponents, the next step according to PEMDAS is to handle multiplication. From the previous step, we have:

• \[5^{2} = 25\]
• \[2^{2} = 4\]

We multiply these results together to get:\[25 \cdot 4 = 100\]
Now that we have executed all multiplication operations, we move on to addition. According to our calculation:

• \[15^{2} = 225\]
• The product from the multiplication: 100

Adding these two results gives us the final value:\[225 + 100 = 325\]This showcases the importance of following the correct steps to achieve an accurate solution.

Step-by-Step Solution Brilliance

A step-by-step approach is a fantastic method for tackling mathematical problems. It helps keep you organized and ensures no details are overlooked. In the case of our problem, breaking it into steps with a focus on each element of PEMDAS:

• First, solve all exponents.
• Next, address multiplication.
• Finally, complete the process with addition.

This practice not only leads to the correct solution but also builds understanding and confidence in mathematical reasoning. By showing the process transparently, it demystifies each mathematical operation, ensuring that anyone learning this approach can replicate it in future exercises. Each step confirms the fundamental rule of mathematics: solving pieces of the puzzle step-by-step ultimately leads to the whole picture.