Simplifying the Given Expression

Oct 13, 2025 by talynt.org 106 views

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves rewriting complex expressions in a simpler form, often by combining like terms or canceling out common factors. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given problem: $\frac{3 x^{-6} y^{-3}}{15 x^2 y^{10}}$ . We will assume that $x \neq 0, y \neq 0$ .

Understanding Exponents

Before we dive into simplifying the given expression, it's essential to understand the concept of exponents. An exponent is a small number that is placed above and to the right of a number or a variable, indicating how many times the base is multiplied by itself. For example, $x^2$ means $x$ multiplied by itself twice, or $x \cdot x$ . Similarly, $x^{-2}$ means $\frac{1}{x^2}$ , or the reciprocal of $x$ squared.

Simplifying the Given Expression

Now that we have a basic understanding of exponents, let's simplify the given expression: $\frac{3 x^{-6} y^{-3}}{15 x^2 y^{10}}$ . To simplify this expression, we will use the following steps:

Combine like terms: We will combine the coefficients (numbers in front of the variables) and the variables with the same exponent.
Cancel out common factors: We will cancel out any common factors in the numerator and denominator.

Step 1: Combine Like Terms

To combine like terms, we will first rewrite the expression with the coefficients and variables separated:

\frac{3 x^{-6} y^{-3}}{15 x^2 y^{10}} = \frac{3}{15} \cdot \frac{x^{-6}}{x^2} \cdot \frac{y^{-3}}{y^{10}}

Next, we will simplify the coefficients and the variables separately:

\frac{3}{15} = \frac{1}{5}

\frac{x^{-6}}{x^2} = x^{-6-2} = x^{-8}

\frac{y^{-3}}{y^{10}} = y^{-3-10} = y^{-13}

Step 2: Cancel Out Common Factors

Now that we have simplified the coefficients and variables, we can cancel out any common factors in the numerator and denominator:

\frac{1}{5} \cdot x^{-8} \cdot y^{-13} = \frac{1}{5} \cdot \frac{1}{x^8} \cdot \frac{1}{y^{13}}

Final Answer

After simplifying the given expression, we get:

\frac{1}{5 x^8 y^{13}}

This is the final answer to the problem.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of exponents and algebraic manipulation. In this article, we have explored the process of simplifying algebraic expressions, with a focus on the given problem: $\frac{3 x^{-6} y^{-3}}{15 x^2 y^{10}}$ . We have used the steps of combining like terms and canceling out common factors to simplify the expression and arrive at the final answer: $\frac{1}{5 x^8 y^{13}}$ .

Discussion

The given problem is a classic example of simplifying algebraic expressions, and it requires a deep understanding of exponents and algebraic manipulation. The problem is designed to test the student's ability to simplify complex expressions and arrive at the final answer.

In this article, we have used the steps of combining like terms and canceling out common factors to simplify the expression. However, there are other methods to simplify algebraic expressions, such as using the distributive property or factoring out common factors.

Additional Resources

For further practice and review, we recommend the following resources:

Khan Academy: Algebraic Expressions
Mathway: Simplifying Algebraic Expressions
Wolfram Alpha: Algebraic Manipulation

Final Thoughts

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given problem: $\frac{3 x^{-6} y^{-3}}{15 x^2 y^{10}}$ . We used the steps of combining like terms and canceling out common factors to simplify the expression and arrive at the final answer: $\frac{1}{5 x^8 y^{13}}$ . In this article, we will provide a Q&A guide to help you better understand the process of simplifying algebraic expressions.

Q&A Guide

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

A: The first step in simplifying an algebraic expression is to combine like terms. This involves combining the coefficients (numbers in front of the variables) and the variables with the same exponent.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you can rewrite the expression with positive exponents by taking the reciprocal of the expression. For example, $x^{-2}$ can be rewritten as $\frac{1}{x^2}$ .

Q: What is the distributive property, and how is it used in simplifying algebraic expressions?

A: The distributive property is a rule that states that a single term can be distributed to multiple terms. For example, $a(b+c)$ can be rewritten as $ab+ac$ . The distributive property is used in simplifying algebraic expressions by distributing a single term to multiple terms.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you can multiply the numerator and denominator by the same value to eliminate the fraction. For example, $\frac{1}{2} \cdot \frac{3}{4}$ can be rewritten as $\frac{3}{8}$ .

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

A: The final step in simplifying an algebraic expression is to cancel out any common factors in the numerator and denominator.

Common Mistakes to Avoid

1. Not combining like terms

Not combining like terms can lead to a more complex expression that is difficult to simplify.

2. Not canceling out common factors

Not canceling out common factors can lead to a more complex expression that is difficult to simplify.

3. Not using the distributive property

Not using the distributive property can lead to a more complex expression that is difficult to simplify.

Tips and Tricks

1. Use the distributive property to simplify expressions

Using the distributive property can help simplify expressions by distributing a single term to multiple terms.

2. Use fractions to simplify expressions

Using fractions can help simplify expressions by eliminating the need for multiple terms.

3. Use negative exponents to simplify expressions

Using negative exponents can help simplify expressions by rewriting the expression with positive exponents.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of exponents and algebraic manipulation. In this article, we have provided a Q&A guide to help you better understand the process of simplifying algebraic expressions. We hope that this article has provided a clear and concise explanation of the process of simplifying algebraic expressions.

Additional Resources

For further practice and review, we recommend the following resources:

Khan Academy: Algebraic Expressions
Mathway: Simplifying Algebraic Expressions
Wolfram Alpha: Algebraic Manipulation