Problem 27 - Algebra 1

Question

Evaluate the expression. Write fractions in simplest form. $$ \left(\frac{6}{2}\right)^{3} $$

Step-by-Step Solution

Answer

The expression \((\frac{6}{2})^{3}\) simplifies to 27.

Step 1: Perform Division

First, execute the division inside the parentheses. \( \frac{6}{2} = 3 \). So, the revised expression is \(3^{3}\).

Step 2: Apply Exponentiation

Now compute the expression \(3^{3}\). This is achieved by multiplying 3 by itself twice, as the exponent tells us how many times to multiply the base (3) by itself. Hence, \(3^{3} = 3 * 3 * 3 = 27\).

Step 3: Simplify Result

The result, 27, cannot be further simplified as the exercise didn’t result in a fraction. Hence, the expression \((\frac{6}{2})^{3}\) evaluates to 27.

Key Concepts

Simplify Fractions

Fractions can feel tricky at first, but simplifying them makes them easier to work with. A fraction is in its simplest form when the numerator (top number) and the denominator (bottom number) have no common divisors other than 1. To simplify a fraction:

• Identify the greatest common factor (GCF) for both the numerator and the denominator.
• Divide both the top and the bottom by this GCF.

In the expression \[\left(\frac{6}{2}\right)\]the GCF of 6 and 2 is 2. By dividing both 6 and 2 by their GCF:

• \[\frac{6}{2} = \frac{3}{1}\]

This gives us the simplest form, which is 3. Simplifying fractions is about making them as straightforward as possible, so always check if the numbers can be divided by a common factor before you move on to further calculations.

Division

Division is one of the four basic operations in arithmetic. It involves splitting a number into equal parts or groups. In mathematical expressions, division shows how many times one number is contained within another. Consider the example provided:\[\left(\frac{6}{2}\right)\]When you perform division here, you're finding out how many times 2 goes into 6.

• 6 divided by 2 equals 3 because when 2 is multiplied by 3, it returns to 6.

Division uses the symbol "/" or a fraction bar, and understanding it is crucial for properly managing mathematical problems that include dividing quantities. Practicing with simple numbers, like dividing 6 by 2, builds confidence for more complex math operations.

Order of Operations

The order of operations is essential for tackling any mathematical expression correctly. It dictates the sequence in which you should solve parts of an expression. The common acronym PEMDAS can help you remember:

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

Let's break down the expression \[\left(\frac{6}{2}\right)^3\]Following PEMDAS:

• Do the operations inside the parentheses first; here, it’s the division \[\frac{6}{2} = 3\].
• Next, deal with the exponent, calculated as \[3^3 = 3 \times 3 \times 3 = 27\].

Adhering to this order avoids mistakes, especially in long and complex expressions. Without it, calculations might yield incorrect results, so always keep PEMDAS in mind!