Perform The Indicated Operation, Then Simplify:$\[ \frac{6b}{10a^2} - \frac{1}{5a} + \frac{2b^2}{a^3} \\]$\[ \frac{30ab - 50 - 100a^3}{a^6} \\]$\[ \frac{a^4 - 50a^6 + 100}{50} \\]$\[ \frac{3ab - A^2 +

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying complex algebraic expressions, using the given expression as a case study. We will break down the expression into smaller parts, perform the indicated operations, and then simplify the resulting expression.

The Given Expression

The given expression is:

6b10a2−15a+2b2a3\frac{6b}{10a^2} - \frac{1}{5a} + \frac{2b^2}{a^3}

This expression consists of three fractions, each with a different denominator. To simplify this expression, we need to find a common denominator and then combine the fractions.

Finding a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the denominators. In this case, the denominators are 10a210a^2, 5a5a, and a3a^3. The LCM of these denominators is 10a510a^5.

Rewriting the Fractions

Now that we have found the common denominator, we can rewrite each fraction with the common denominator.

6b10a2=6b⋅a310a2⋅a3=6ba310a5\frac{6b}{10a^2} = \frac{6b \cdot a^3}{10a^2 \cdot a^3} = \frac{6ba^3}{10a^5}

15a=1⋅a45a⋅a4=a45a5\frac{1}{5a} = \frac{1 \cdot a^4}{5a \cdot a^4} = \frac{a^4}{5a^5}

2b2a3=2b2⋅10a2a3⋅10a2=20b2a210a5\frac{2b^2}{a^3} = \frac{2b^2 \cdot 10a^2}{a^3 \cdot 10a^2} = \frac{20b^2a^2}{10a^5}

Combining the Fractions

Now that we have rewritten each fraction with the common denominator, we can combine them.

6ba310a5−a45a5+20b2a210a5\frac{6ba^3}{10a^5} - \frac{a^4}{5a^5} + \frac{20b^2a^2}{10a^5}

To combine the fractions, we need to find a common numerator. In this case, the common numerator is 6ba3−a4+20b2a26ba^3 - a^4 + 20b^2a^2.

Simplifying the Expression

Now that we have combined the fractions, we can simplify the expression.

6ba3−a4+20b2a210a5\frac{6ba^3 - a^4 + 20b^2a^2}{10a^5}

To simplify this expression, we can factor out the common terms.

a2(6b−a2+20b2)10a5\frac{a^2(6b - a^2 + 20b^2)}{10a^5}

The Second Expression

The second expression is:

30ab−50−100a3a6\frac{30ab - 50 - 100a^3}{a^6}

This expression consists of a single fraction with a polynomial numerator and a power of aa in the denominator. To simplify this expression, we need to factor the numerator and then cancel out any common factors.

Factoring the Numerator

The numerator of the fraction is 30ab−50−100a330ab - 50 - 100a^3. We can factor out the common terms.

30ab−50−100a3=10a(3b−5−10a2)30ab - 50 - 100a^3 = 10a(3b - 5 - 10a^2)

Simplifying the Expression

Now that we have factored the numerator, we can simplify the expression.

10a(3b−5−10a2)a6\frac{10a(3b - 5 - 10a^2)}{a^6}

To simplify this expression, we can cancel out the common factor of aa.

10(3b−5−10a2)a5\frac{10(3b - 5 - 10a^2)}{a^5}

The Third Expression

The third expression is:

a4−50a6+10050\frac{a^4 - 50a^6 + 100}{50}

This expression consists of a single fraction with a polynomial numerator and a constant denominator. To simplify this expression, we need to factor the numerator and then cancel out any common factors.

Factoring the Numerator

The numerator of the fraction is a4−50a6+100a^4 - 50a^6 + 100. We can factor out the common terms.

a4−50a6+100=(a2−50a4+100)a^4 - 50a^6 + 100 = (a^2 - 50a^4 + 100)

Simplifying the Expression

Now that we have factored the numerator, we can simplify the expression.

(a2−50a4+100)50\frac{(a^2 - 50a^4 + 100)}{50}

To simplify this expression, we can cancel out the common factor of 5050.

a2−50a4+10050=a250−a4+2a4\frac{a^2 - 50a^4 + 100}{50} = \frac{a^2}{50} - a^4 + \frac{2}{a^4}

The Final Expression

The final expression is:

3ab−a2+2a450\frac{3ab - a^2 + \frac{2}{a^4}}{50}

This expression consists of a single fraction with a polynomial numerator and a constant denominator. To simplify this expression, we need to factor the numerator and then cancel out any common factors.

Conclusion

In this article, we have explored the process of simplifying complex algebraic expressions. We have used the given expression as a case study and broken it down into smaller parts. We have found a common denominator, rewritten each fraction with the common denominator, combined the fractions, and then simplified the resulting expression. We have also factored the numerator and canceled out any common factors to simplify the expression. The final expression is a simplified version of the original expression, and it can be used as a starting point for further calculations.

Discussion

The process of simplifying complex algebraic expressions is an important skill for students and professionals alike. It requires a deep understanding of algebraic concepts and the ability to break down complex expressions into smaller parts. In this article, we have used the given expression as a case study and demonstrated the process of simplifying complex algebraic expressions. We have also provided a step-by-step guide to simplifying complex algebraic expressions, which can be used as a reference for future calculations.

References

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

A: The first step in simplifying a complex algebraic expression is to identify the common denominator of the fractions. This is the least common multiple (LCM) of the denominators.

Q: How do I find the common denominator of a complex algebraic expression?

A: To find the common denominator, you need to identify the least common multiple (LCM) of the denominators. This can be done by listing the multiples of each denominator and finding the smallest multiple that is common to all of them.

Q: What is the next step in simplifying a complex algebraic expression?

A: Once you have found the common denominator, you need to rewrite each fraction with the common denominator. This involves multiplying the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.

Q: How do I combine fractions with different denominators?

A: To combine fractions with different denominators, you need to find a common denominator and then add or subtract the numerators. This can be done by rewriting each fraction with the common denominator and then adding or subtracting the numerators.

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

A: The final step in simplifying a complex algebraic expression is to simplify the resulting expression. This involves combining like terms, canceling out any common factors, and rewriting the expression in its simplest form.

Q: How do I simplify a complex algebraic expression with multiple variables?

A: To simplify a complex algebraic expression with multiple variables, you need to follow the same steps as before. However, you will need to take into account the relationships between the variables and simplify the expression accordingly.

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

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

  • Failing to identify the common denominator
  • Not rewriting each fraction with the common denominator
  • Not combining like terms
  • Not canceling out any common factors
  • Not rewriting the expression in its simplest form

Q: How can I practice simplifying complex algebraic expressions?

A: You can practice simplifying complex algebraic expressions by working through examples and exercises. You can also use online resources and practice problems to help you build your skills.

Q: What are some real-world applications of simplifying complex algebraic expressions?

A: Simplifying complex algebraic expressions has many real-world applications, including:

  • Physics and engineering: Simplifying complex algebraic expressions is essential in physics and engineering, where complex equations are used to model real-world phenomena.
  • Computer science: Simplifying complex algebraic expressions is also important in computer science, where complex algorithms are used to solve problems.
  • Economics: Simplifying complex algebraic expressions is also used in economics, where complex models are used to analyze economic data.

Q: How can I use technology to simplify complex algebraic expressions?

A: You can use technology, such as calculators and computer software, to simplify complex algebraic expressions. These tools can help you to identify the common denominator, rewrite each fraction with the common denominator, and simplify the resulting expression.

Q: What are some tips for simplifying complex algebraic expressions?

A: Some tips for simplifying complex algebraic expressions include:

  • Take your time and work through the problem carefully
  • Identify the common denominator and rewrite each fraction with the common denominator
  • Combine like terms and cancel out any common factors
  • Rewrite the expression in its simplest form
  • Use technology, such as calculators and computer software, to help you simplify the expression.

Conclusion

Simplifying complex algebraic expressions is an important skill that has many real-world applications. By following the steps outlined in this article, you can simplify complex algebraic expressions and build your skills in algebra. Remember to take your time, work through the problem carefully, and use technology, such as calculators and computer software, to help you simplify the expression.