Simplify The Following Expression:$[ \frac \left(2-\frac{1}{5}\right) 2}{\left(3-\frac{2}{9}\right) {-1}} \div \frac{\left(\frac{6}{7} \cdot \frac{5}{4}-\frac{2}{7} \frac{1 {2}\right)^3}{\left(\frac{1}{2}-\frac{1}{3} \cdot \frac{1}{4} :

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Introduction

Mathematical expressions can be complex and challenging to simplify. In this article, we will focus on simplifying a given mathematical expression that involves fractions, exponents, and arithmetic operations. The expression is as follows:

(2−15)2(3−29)−1÷(67⋅54−27:12)3(12−13⋅14:15)−2{ \frac{\left(2-\frac{1}{5}\right)^2}{\left(3-\frac{2}{9}\right)^{-1}} \div \frac{\left(\frac{6}{7} \cdot \frac{5}{4}-\frac{2}{7} : \frac{1}{2}\right)^3}{\left(\frac{1}{2}-\frac{1}{3} \cdot \frac{1}{4} : \frac{1}{5}\right)^{-2}} }

Breaking Down the Expression

To simplify the given expression, we need to break it down into smaller parts and simplify each part separately. Let's start by simplifying the fractions within the parentheses.

Simplifying the Fractions

The expression contains several fractions that need to be simplified. Let's start by simplifying the fractions within the parentheses.

  • (2−15)\left(2-\frac{1}{5}\right) can be simplified as follows:

    2−15=105−15=95{ 2-\frac{1}{5} = \frac{10}{5} - \frac{1}{5} = \frac{9}{5} }

  • (3−29)\left(3-\frac{2}{9}\right) can be simplified as follows:

    3−29=279−29=259{ 3-\frac{2}{9} = \frac{27}{9} - \frac{2}{9} = \frac{25}{9} }

  • (67⋅54−27:12)\left(\frac{6}{7} \cdot \frac{5}{4}-\frac{2}{7} : \frac{1}{2}\right) can be simplified as follows:

    67⋅54=3028=1514{ \frac{6}{7} \cdot \frac{5}{4} = \frac{30}{28} = \frac{15}{14} }

    1514−27=1514−414=1114{ \frac{15}{14} - \frac{2}{7} = \frac{15}{14} - \frac{4}{14} = \frac{11}{14} }

    1114:12=1114⋅21=2214=117{ \frac{11}{14} : \frac{1}{2} = \frac{11}{14} \cdot \frac{2}{1} = \frac{22}{14} = \frac{11}{7} }

  • (12−13⋅14:15)\left(\frac{1}{2}-\frac{1}{3} \cdot \frac{1}{4} : \frac{1}{5}\right) can be simplified as follows:

    13⋅14=112{ \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12} }

    12−112=612−112=512{ \frac{1}{2} - \frac{1}{12} = \frac{6}{12} - \frac{1}{12} = \frac{5}{12} }

    512:15=512⋅51=2512{ \frac{5}{12} : \frac{1}{5} = \frac{5}{12} \cdot \frac{5}{1} = \frac{25}{12} }

Simplifying the Expression

Now that we have simplified the fractions within the parentheses, let's simplify the expression.

Simplifying the Exponents

The expression contains exponents that need to be simplified. Let's start by simplifying the exponents.

  • (2−15)2\left(2-\frac{1}{5}\right)^2 can be simplified as follows:

    (95)2=8125{ \left(\frac{9}{5}\right)^2 = \frac{81}{25} }

  • (3−29)−1\left(3-\frac{2}{9}\right)^{-1} can be simplified as follows:

    (259)−1=925{ \left(\frac{25}{9}\right)^{-1} = \frac{9}{25} }

  • (67⋅54−27:12)3\left(\frac{6}{7} \cdot \frac{5}{4}-\frac{2}{7} : \frac{1}{2}\right)^3 can be simplified as follows:

    (117)3=1331343{ \left(\frac{11}{7}\right)^3 = \frac{1331}{343} }

  • (12−13⋅14:15)−2\left(\frac{1}{2}-\frac{1}{3} \cdot \frac{1}{4} : \frac{1}{5}\right)^{-2} can be simplified as follows:

    (2512)−2=144625{ \left(\frac{25}{12}\right)^{-2} = \frac{144}{625} }

Simplifying the Expression

Now that we have simplified the exponents, let's simplify the expression.

(2−15)2(3−29)−1÷(67⋅54−27:12)3(12−13⋅14:15)−2{ \frac{\left(2-\frac{1}{5}\right)^2}{\left(3-\frac{2}{9}\right)^{-1}} \div \frac{\left(\frac{6}{7} \cdot \frac{5}{4}-\frac{2}{7} : \frac{1}{2}\right)^3}{\left(\frac{1}{2}-\frac{1}{3} \cdot \frac{1}{4} : \frac{1}{5}\right)^{-2}} }

can be simplified as follows:

8125925÷1331343144625{ \frac{\frac{81}{25}}{\frac{9}{25}} \div \frac{\frac{1331}{343}}{\frac{144}{625}} }

8125⋅259÷1331343⋅625144{ \frac{81}{25} \cdot \frac{25}{9} \div \frac{1331}{343} \cdot \frac{625}{144} }

819÷1331⋅625343⋅144{ \frac{81}{9} \div \frac{1331 \cdot 625}{343 \cdot 144} }

9÷82922549152{ 9 \div \frac{829225}{49152} }

9⋅49152829225{ \frac{9 \cdot 49152}{829225} }

441888829225{ \frac{441888}{829225} }

Conclusion

In this article, we simplified a complex mathematical expression that involved fractions, exponents, and arithmetic operations. We broke down the expression into smaller parts, simplified each part separately, and then combined the simplified parts to simplify the expression. The final simplified expression is 441888829225\frac{441888}{829225}.

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Introduction

In our previous article, we simplified a complex mathematical expression that involved fractions, exponents, and arithmetic operations. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the main concept behind simplifying the given expression?

A: The main concept behind simplifying the given expression is to break it down into smaller parts, simplify each part separately, and then combine the simplified parts to simplify the expression.

Q: What are the steps involved in simplifying the expression?

A: The steps involved in simplifying the expression are:

  1. Simplify the fractions within the parentheses.
  2. Simplify the exponents.
  3. Combine the simplified parts to simplify the expression.

Q: How do you simplify fractions within the parentheses?

A: To simplify fractions within the parentheses, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponents (such as squaring or cubing).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: How do you simplify exponents?

A: To simplify exponents, you need to follow the rules of exponents:

  1. When multiplying two numbers with the same base, add the exponents.
  2. When dividing two numbers with the same base, subtract the exponents.
  3. When raising a power to a power, multiply the exponents.

Q: What is the final simplified expression?

A: The final simplified expression is 441888829225\frac{441888}{829225}.

Q: How do you evaluate the expression?

A: To evaluate the expression, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponents (such as squaring or cubing).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

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

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

  1. Not following the order of operations (PEMDAS).
  2. Not simplifying fractions within the parentheses.
  3. Not simplifying exponents.
  4. Not combining the simplified parts to simplify the expression.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given expression. We covered the main concept behind simplifying the expression, the steps involved in simplifying the expression, and some common mistakes to avoid when simplifying expressions. We hope this article has been helpful in understanding the simplification of the given expression.

Additional Resources

For more information on simplifying expressions, you can refer to the following resources:

  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

Final Thoughts

Simplifying expressions is an important skill in mathematics that can be used to solve a wide range of problems. By following the steps outlined in this article, you can simplify complex expressions and arrive at the final answer. Remember to always follow the order of operations (PEMDAS) and to simplify fractions within the parentheses and exponents. With practice and patience, you can become proficient in simplifying expressions and solving complex mathematical problems.