Simplify The Expression Completely: (26 X¹⁹ Y²⁰) / (34 X⁷ Y⁶)
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Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying the given expression: (26 x¹⁹ y²⁰) / (34 x⁷ y⁶). We will break down the process into manageable steps, making it easy to understand and follow.
Understanding the Expression
Before we dive into simplifying the expression, let's take a closer look at it. The given expression is a fraction, where the numerator is (26 x¹⁹ y²⁰) and the denominator is (34 x⁷ y⁶). To simplify this expression, we need to apply the rules of exponents and fractions.
Simplifying the Numerator and Denominator
Let's start by simplifying the numerator and denominator separately.
Simplifying the Numerator
The numerator is (26 x¹⁹ y²⁰). We can simplify this by breaking it down into its prime factors. The number 26 can be expressed as 2 × 13, and the exponent 19 can be written as 19. The variable y has an exponent of 20.
numerator = 2 * 13 * (x ** 19) * (y ** 20)
Simplifying the Denominator
The denominator is (34 x⁷ y⁶). We can simplify this by breaking it down into its prime factors. The number 34 can be expressed as 2 × 17, and the exponent 7 can be written as 7. The variable y has an exponent of 6.
denominator = 2 * 17 * (x ** 7) * (y ** 6)
Canceling Out Common Factors
Now that we have simplified the numerator and denominator, we can cancel out any common factors. In this case, we can cancel out the common factor of 2, x, and y.
simplified_expression = (13 * (x ** 12) * (y ** 14)) / (17 * (x ** 7) * (y ** 6))
Applying the Quotient of Powers Rule
The quotient of powers rule states that when we divide two powers with the same base, we can subtract the exponents. In this case, we can apply the quotient of powers rule to simplify the expression further.
simplified_expression = (13 * x ** (12 - 7) * y ** (14 - 6)) / 17
Simplifying the Expression Further
Now that we have applied the quotient of powers rule, we can simplify the expression further by evaluating the exponents.
simplified_expression = (13 * x ** 5 * y ** 8) / 17
Conclusion
In conclusion, simplifying the given expression (26 x¹⁹ y²⁰) / (34 x⁷ y⁶) requires a step-by-step approach. We started by simplifying the numerator and denominator separately, then canceled out common factors, and finally applied the quotient of powers rule to simplify the expression further. The final simplified expression is (13 * x ** 5 * y ** 8) / 17.
Frequently Asked Questions
Q: What is the quotient of powers rule?
A: The quotient of powers rule states that when we divide two powers with the same base, we can subtract the exponents.
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, you need to apply the rules of exponents and fractions. Start by simplifying the numerator and denominator separately, then cancel out common factors, and finally apply the quotient of powers rule.
Q: What is the final simplified expression?
A: The final simplified expression is (13 * x ** 5 * y ** 8) / 17.
Additional Resources
For more information on simplifying algebraic expressions, check out the following resources:
- Khan Academy: Simplifying Algebraic Expressions
- Mathway: Simplifying Algebraic Expressions
- Wolfram Alpha: Simplifying Algebraic Expressions
Final Thoughts
Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article, you can simplify even the most complex expressions. Remember to always start by simplifying the numerator and denominator separately, then cancel out common factors, and finally apply the quotient of powers rule. With practice and patience, you will become a master of simplifying algebraic expressions.
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Introduction
Simplifying algebraic expressions is a crucial skill for students and professionals alike. In our previous article, we provided a step-by-step guide on how to simplify the expression (26 x¹⁹ y²⁰) / (34 x⁷ y⁶). However, we understand that sometimes, you may have questions or need further clarification on certain concepts. That's why we've put together this Q&A guide to help you better understand algebraic expression simplification.
Q&A
Q: What is the first step in simplifying an algebraic expression?
A: The first step in simplifying an algebraic expression is to simplify the numerator and denominator separately. This involves breaking down the expression into its prime factors and simplifying each part.
Q: How do I simplify the numerator and denominator?
A: To simplify the numerator and denominator, you need to break down the expression into its prime factors. For example, if the numerator is (26 x¹⁹ y²⁰), you can break it down into its prime factors as 2 × 13 × (x¹⁹) × (y²⁰).
Q: What is the quotient of powers rule?
A: The quotient of powers rule states that when you divide two powers with the same base, you can subtract the exponents. For example, if you have (x⁸) / (x³), you can simplify it to x⁸-³, which is x⁵.
Q: How do I apply the quotient of powers rule?
A: To apply the quotient of powers rule, you need to identify the common base and subtract the exponents. For example, if you have (x⁸ y⁴) / (x³ y²), you can simplify it to (x⁸-³) × (y⁴-²), which is x⁵ y².
Q: What is the final simplified expression?
A: The final simplified expression depends on the original expression. In our previous article, we simplified the expression (26 x¹⁹ y²⁰) / (34 x⁷ y⁶) to (13 * x ** 5 * y ** 8) / 17.
Q: How do I know when to simplify an algebraic expression?
A: You should simplify an algebraic expression when it is necessary to make the expression more manageable or to solve a problem. Simplifying an expression can help you to:
- Make calculations easier
- Solve problems more efficiently
- Understand complex concepts better
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Not simplifying the numerator and denominator separately
- Not applying the quotient of powers rule correctly
- Not canceling out common factors
- Not checking for errors in the final simplified expression
Additional Tips and Resources
Tip 1: Practice, Practice, Practice
Simplifying algebraic expressions requires practice to become proficient. Try simplifying different expressions to build your skills and confidence.
Tip 2: Use Online Resources
There are many online resources available to help you simplify algebraic expressions, including calculators, worksheets, and video tutorials.
Tip : Seek Help When Needed
Don't be afraid to ask for help if you're struggling to simplify an algebraic expression. Reach out to a teacher, tutor, or classmate for assistance.
Conclusion
Simplifying algebraic expressions is a crucial skill for students and professionals alike. By following the steps outlined in this Q&A guide, you can simplify even the most complex expressions. Remember to always start by simplifying the numerator and denominator separately, then apply the quotient of powers rule, and finally check for errors in the final simplified expression. With practice and patience, you will become a master of simplifying algebraic expressions.
Frequently Asked Questions
Q: What is the best way to learn algebraic expression simplification?
A: The best way to learn algebraic expression simplification is through practice and patience. Try simplifying different expressions to build your skills and confidence.
Q: How do I know if I've simplified an expression correctly?
A: To check if you've simplified an expression correctly, make sure to:
- Simplify the numerator and denominator separately
- Apply the quotient of powers rule correctly
- Cancel out common factors
- Check for errors in the final simplified expression
Q: What are some real-world applications of algebraic expression simplification?
A: Algebraic expression simplification has many real-world applications, including:
- Physics and engineering
- Computer science and programming
- Economics and finance
- Data analysis and statistics
Additional Resources
For more information on algebraic expression simplification, check out the following resources:
- Khan Academy: Algebraic Expression Simplification
- Mathway: Algebraic Expression Simplification
- Wolfram Alpha: Algebraic Expression Simplification