Problem 28 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(37-1 \cdot 6^{2}\)

Step-by-Step Solution

Answer

The value is 1.

Step 1: Identify the Order of Operations

To solve the expression \(37 - 1 \cdot 6^{2}\), use the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this expression, we have exponentiation, multiplication, and subtraction to consider.

Step 2: Solve the Exponentiation

Calculate the exponent first. The expression has \(6^{2}\), which means \(6\times6\). Solving this gives us \(6^{2} = 36\).

Step 3: Perform the Multiplication

Next, perform the multiplication. The expression now is \(1 \cdot 36\). Calculating this gives \(1 \cdot 36 = 36\).

Step 4: Execute the Subtraction

Finally, complete the subtraction in the expression. Substitute the result of the previous multiplication into the expression: \(37 - 36\). Solving this gives \(37 - 36 = 1\).

Key Concepts

PEMDAS

The order of operations is crucial in solving mathematical expressions and is remembered with the acronym PEMDAS. This sequence helps you decide which operations to perform first. Let's break it down:

• **P**arentheses: Solve anything inside parentheses first.
• **E**xponents: Calculate powers and roots next.
• **M**ultiplication and **D**ivision: Work these from left to right.
• **A**ddition and **S**ubtraction: Handle these last, also from left to right.

For the exercise, "37 minus 1 times 6 squared," the order guided us to first address the exponent, then multiplication, and finally subtraction. Ignoring PEMDAS may lead to incorrect solutions, so it's important to stick to this order.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. It is typically written as a small number (the exponent) to the top right of a base number. For instance, in the expression \(6^2\), 6 is the base, and 2 is the exponent. This signifies \(6 \times 6\), which equals 36.
When solving expressions, exponents take priority right after parentheses (if any). In our exercise, we calculated \(6^2\) first, which simplified the expression significantly. Always remember to tackle exponents before moving on to other operations.

Multiplication

Multiplication is one of the fundamental operations in mathematics. It involves finding the total when one number is taken several times. In our context, after calculating the exponent, we moved on to multiplication.
The expression "1 times 36" simplified to 36 because multiplying any number by 1 results in the same number. Multiplication usually follows after exponents in PEMDAS, so it's crucial to solve any exponentiation before multiplying numbers in an expression.

Subtraction

Subtraction is about taking away or deducting numbers. In mathematical expressions, subtraction is generally one of the last operations due to PEMDAS's rules.
In the exercise "37 minus 36," it was our final step. Once the calculations involving exponents and multiplication were complete, this left us with a simple subtraction task, yielding a result of 1. Always remember to leave subtraction until the operations higher in the hierarchy are finished to ensure an accurate outcome.