Calculate The Expression:${ \frac{3^2}{6} - \frac{23}{6} }$

Introduction

In mathematics, expressions can be complex and require careful simplification to arrive at a solution. One such expression is given as: $\frac{3^2}{6} - \frac{23}{6}$. In this article, we will break down the steps to simplify this expression and provide a clear understanding of the process.

Understanding the Expression

The given expression is a combination of two fractions: $\frac{3^2}{6}$ and $\frac{23}{6}$. To simplify this expression, we need to first evaluate the numerator of the first fraction, which is $3^2$. The exponent $2$ indicates that we need to multiply $3$ by itself.

Evaluating Exponents

The exponent $2$ in the expression $3^2$ means that we need to multiply $3$ by itself: $3^2 = 3 \times 3 = 9$. Therefore, the first fraction becomes $\frac{9}{6}$.

Simplifying the First Fraction

The fraction $\frac{9}{6}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of $9$ and $6$ is $3$. Dividing both the numerator and the denominator by $3$, we get: $\frac{9}{6} = \frac{3}{2}$.

Subtracting the Second Fraction

Now that we have simplified the first fraction, we can subtract the second fraction: $\frac{3}{2} - \frac{23}{6}$. To subtract these fractions, we need to have a common denominator. The least common multiple (LCM) of $2$ and $6$ is $6$. Therefore, we can rewrite the first fraction with a denominator of $6$: $\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}$.

Finding a Common Denominator

Now that both fractions have a common denominator of $6$, we can subtract them: $\frac{9}{6} - \frac{23}{6}$. To subtract these fractions, we need to subtract the numerators while keeping the denominator the same.

Subtracting Numerators

The numerators of the two fractions are $9$ and $23$. Subtracting $9$ from $23$, we get: $23 - 9 = 14$. Therefore, the expression becomes: $\frac{14}{6}$.

Simplifying the Final Fraction

The fraction $\frac{14}{6}$ can be simplified by dividing both the numerator and the denominator by their GCD. The GCD of $14$ and $6$ is $2$. Dividing both the numerator and the denominator by $2$, we get: $\frac{14}{6} = \frac{7}{3}$.

Conclusion

In this article, we simplified the expression $\frac{3^2}{6} - \frac{23}{6}$ by following a step-by-step approach. We evaluated the exponent, simplified the first fraction, subtracted the second fraction, and finally simplified the resulting fraction. The final answer is $\frac{7}{3}$.

Final Answer

$\boxed{\frac{7}{3}}$

Additional Tips and Tricks

• When simplifying complex expressions, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• To simplify fractions, find the greatest common divisor (GCD) of the numerator and the denominator and divide both by the GCD.
• When subtracting fractions, find a common denominator and subtract the numerators while keeping the denominator the same.

Introduction

In our previous article, we simplified the expression $\frac{3^2}{6} - \frac{23}{6}$ by following a step-by-step approach. However, we understand that simplifying complex expressions can be a challenging task, and many of you may have questions about the process. In this article, we will address some of the most frequently asked questions about simplifying complex expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying complex expressions. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate any exponential expressions next (e.g., $2^3$).
• Multiplication and Division: Evaluate any multiplication and division operations from left to right.
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify fractions?

A: To simplify fractions, follow these steps:

1. Find the greatest common divisor (GCD) of the numerator and the denominator.
2. Divide both the numerator and the denominator by the GCD.
3. The resulting fraction is the simplified form.

Q: What is the difference between a numerator and a denominator?

A: In a fraction, the numerator is the number on top, and the denominator is the number on the bottom. For example, in the fraction $\frac{3}{4}$, $3$ is the numerator, and $4$ is the denominator.

Q: How do I subtract fractions?

A: To subtract fractions, follow these steps:

1. Find a common denominator for the fractions.
2. Rewrite each fraction with the common denominator.
3. Subtract the numerators while keeping the denominator the same.
4. Simplify the resulting fraction.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of $2$ and $6$ is $6$, because $6$ is the smallest number that both $2$ and $6$ can divide into evenly.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, follow these steps:

1. List the multiples of each number.
2. Find the smallest multiple that appears in both lists.
3. The LCM is the smallest multiple that appears in both lists.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that two or more numbers have in common. For example, the GCD of $12$ and $18$ is $6$, because $6$ is the largest number that both $12$ and $18$ can divide into evenly.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, follow these steps:

1. List the factors of each number.
2. Find the largest factor that appears in both lists.
3. The GCD is the largest factor that appears in both lists.

Conclusion

Simplifying complex expressions can be a challenging task, but by following the order of operations (PEMDAS) and understanding the concepts of fractions, numerators, denominators, LCM, and GCD, you can simplify even the most complex expressions. We hope this Q&A guide has been helpful in addressing some of the most frequently asked questions about simplifying complex expressions.

Additional Resources

• For more information on simplifying complex expressions, check out our previous article: Simplifying Complex Expressions: A Step-by-Step Guide.
• For a comprehensive guide to fractions, numerators, denominators, LCM, and GCD, check out our article: Understanding Fractions: A Comprehensive Guide.
• For practice problems and exercises, check out our math practice section: Math Practice.