Problem 34 - Algebra 1

Question

Evaluate the expression. $$\left[10+\left(5^{2} \cdot 2\right)\right] \div 6$$

Step-by-Step Solution

Answer

The evaluated result of the expression \(\left[10+\left(5^{2} \cdot 2\right)\right] \div 6\) is 10.

Step 1: Solve the Parentheses

First, perform the operations within parentheses. This involves calculating the exponent and then the multiplication in the expression \((5^{2} \cdot 2)\). The exponent, \(5^{2}\), equates to 25. Thus, the expression becomes \( (25 \cdot 2)\), which equals 50.

Step 2: Solve the Bracketed Expression

Next, substitute the result from the parentheses back into the original expression, making the expression now \((10 + 50) \div 6\). The operation within the brackets needs to be performed next. Adding 10 and 50, yields 60.

Step 3: Perform the Division

Now perform the remaining operation, which is the division of 60 by 6. This gives the final result of 10.

Key Concepts

Order of Operations

Understanding the order of operations is crucial for correctly evaluating mathematical expressions. It dictates the sequence in which operations should be performed to arrive at the right answer.

Here is the standard order to follow:

• Parentheses: solve expressions inside parentheses and other grouping symbols first.
• Exponents: next, calculate powers and roots.
• Multiplication and Division: proceed from left to right, performing all multiplication and division as they appear.
• Addition and Subtraction: finally, handle addition and subtraction from left to right.

An easy way to remember this is the acronym PEMDAS ('Please Excuse My Dear Aunt Sally'), representing Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction.

In our example, we followed these steps. We started inside the parentheses by dealing with the exponent, followed by multiplication, then addressing the addition within the brackets, and finally performing the division.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number. An exponent is written as a small number (the exponent) to the upper right of a base number. It tells us how many times to multiply the base by itself.

For instance, the expression \(\text{5}^{2}\) means 5 multiplied by itself 2 times, which is equivalent to \(\text{5} \times \text{5}\), or 25. When an exponent is present, it's one of the first operations you should look for after dealing with any parentheses.

In the initial expression \(\text{5}^{2} \times 2\), the expression \(\text{5}^{2}\) is calculated first, as per the order of operations, before multiplying by 2. By understanding exponents, you can simplify parts of expressions and make them easier to manage, just as we simplified \(\text{5}^{2}\) to 25 in our first step of the example.

Parentheses in Math

In mathematics, parentheses are used to group parts of an expression that should be evaluated first. Whenever you see an expression within parentheses, you need to solve it before you do anything outside the parentheses.

For example, in the expression \(\text{10} + (\text{5}^{2} \times 2)\), the parentheses around \(\text{5}^{2} \times 2\) tell you to focus on this part first. We calculate the exponent within this sub-expression, as explained in the section on exponents, and then perform any additional operations, such as multiplication.

After solving the innermost parts, we can then proceed to anything outside the parentheses. If there are multiple levels of grouping symbols—such as brackets, braces, and parentheses—start with the innermost one. This step-by-step procedure ensures that complex expressions are simplified in a structured manner, avoiding mistakes that could arise from random calculation sequences.