Simplify The Expression: 99 A 6 B 7 C 33 A B 2 \frac{99 A^6 B^7 C}{33 A B^2} 33 A B 2 99 A 6 B 7 C

Mar 10, 2025 by ADMIN 101 views

Simplify the Expression: A Comprehensive Guide to Algebraic Manipulation

Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving complex equations and problems. It involves reducing a given expression to its simplest form by combining like terms, canceling out common factors, and performing other mathematical operations. In this article, we will focus on simplifying the expression $\frac{99 a^6 b^7 c}{33 a b^2}$ , which requires a deep understanding of algebraic manipulation and factorization.

Understanding the Expression

The given expression is a fraction with a numerator and a denominator. The numerator is $99 a^6 b^7 c$ , and the denominator is $33 a b^2$ . To simplify this expression, we need to analyze the numerator and the denominator separately and then combine them.

Breaking Down the Numerator

The numerator is $99 a^6 b^7 c$ . We can break it down into its prime factors to simplify it further. The prime factorization of 99 is $3^2 \times 11$ , and the prime factorization of $a^6$ is $a^6$ , the prime factorization of $b^7$ is $b^7$ , and the prime factorization of $c$ is $c$ . Therefore, the numerator can be written as $3^2 \times 11 \times a^6 \times b^7 \times c$ .

Breaking Down the Denominator

The denominator is $33 a b^2$ . We can break it down into its prime factors to simplify it further. The prime factorization of 33 is $3 \times 11$ , and the prime factorization of $a$ is $a$ , the prime factorization of $b^2$ is $b^2$ . Therefore, the denominator can be written as $3 \times 11 \times a \times b^2$ .

Simplifying the Expression

Now that we have broken down the numerator and the denominator, we can simplify the expression by canceling out common factors. The numerator and the denominator have common factors of $3$ , $11$ , $a$ , and $b^2$ . We can cancel out these common factors to simplify the expression.

Canceling Out Common Factors

To cancel out common factors, we need to divide the numerator and the denominator by the common factors. In this case, we can divide the numerator and the denominator by $3$ , $11$ , $a$ , and $b^2$ . This will give us a simplified expression.

Simplified Expression

After canceling out common factors, the simplified expression is $\frac{3^1 \times a^5 \times b^5 \times c}{1}$ . This is the simplest form of the given expression.

Conclusion

Simplifying the expression $\frac{99 a^6 b^7 c}{33 a b^2}$ requires a deep understanding of algebraic manipulation and factorization. By breaking down the numerator and the denominator into their prime factors and canceling out common factors, we can simplify the expression to its simplest form. This skill is essential in solving complex equations and problems in algebra.

Tips and Tricks

Always break down the numerator and the denominator into their prime factors to simplify the expression.
Cancel out common factors to simplify the expression.
Use the distributive property to simplify the expression.
Use the commutative property to simplify the expression.
Use the associative property to simplify the expression.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

Physics: Simplifying expressions is essential in solving complex equations and problems in physics.
Engineering: Simplifying expressions is essential in solving complex equations and problems in engineering.
Computer Science: Simplifying expressions is essential in solving complex equations and problems in computer science.
Economics: Simplifying expressions is essential in solving complex equations and problems in economics.

Final Thoughts

Simplifying expressions is a crucial skill that helps in solving complex equations and problems in algebra. By breaking down the numerator and the denominator into their prime factors and canceling out common factors, we can simplify the expression to its simplest form. This skill is essential in many real-world applications, including physics, engineering, computer science, and economics.

Introduction

In our previous article, we discussed how to simplify the expression $\frac{99 a^6 b^7 c}{33 a b^2}$ . We broke down the numerator and the denominator into their prime factors and canceled out common factors to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to break down the numerator and the denominator into their prime factors.

Q: How do I break down the numerator and the denominator into their prime factors?

A: To break down the numerator and the denominator into their prime factors, you need to find the prime factorization of each number and variable. For example, the prime factorization of 99 is $3^2 \times 11$ , and the prime factorization of $a^6$ is $a^6$ .

Q: What is the next step in simplifying an expression?

A: The next step in simplifying an expression is to cancel out common factors. To cancel out common factors, you need to divide the numerator and the denominator by the common factors.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to divide the numerator and the denominator by the common factors. For example, if the numerator and the denominator have a common factor of $a$ , you can cancel out this factor by dividing the numerator and the denominator by $a$ .

Q: What is the final step in simplifying an expression?

A: The final step in simplifying an expression is to simplify the resulting expression. This may involve combining like terms, canceling out any remaining common factors, and performing other mathematical operations.

Q: How do I know if an expression is simplified?

A: An expression is simplified when it cannot be simplified further. This means that there are no more common factors to cancel out, and the expression cannot be combined or rearranged in any way.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

Not breaking down the numerator and the denominator into their prime factors
Not canceling out common factors
Not simplifying the resulting expression
Making errors when dividing or multiplying

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through examples and exercises. You can also try simplifying expressions on your own and then checking your work to see if you made any mistakes.

Tips and Tricks

Always break down the numerator and the denominator into their prime factors to simplify the expression.
Cancel out common factors to simplify the expression.
Use the distributive property to simplify the expression.
Use the commutative property to simplify the expression.
Use the associative property to simplify the expression.

Real-World Applications

Simplifying expressions is a crucial skill in many real-world applications, including:

Physics: Simplifying expressions is essential in solving complex equations and problems in physics.
Engineering: Simplifying expressions is essential in solving complex equations and problems in engineering.
Computer Science: Simplifying expressions is essential in solving complex equations and problems in computer science.
Economics: Simplifying expressions is essential in solving complex equations and problems in economics.