Solve The Equation:${ \frac{2 {-1}}{2 {-1} + 3^{-1}} + \frac{2 {-1}}{2 {-1} - 3^{-1}} = \frac{18}{5} }$

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Introduction


In this article, we will delve into the world of mathematics and explore a complex equation involving fractions and exponents. The equation in question is 2βˆ’12βˆ’1+3βˆ’1+2βˆ’12βˆ’1βˆ’3βˆ’1=185\frac{2^{-1}}{2^{-1} + 3^{-1}} + \frac{2^{-1}}{2^{-1} - 3^{-1}} = \frac{18}{5}. Our goal is to simplify and solve this equation, providing a clear and concise explanation of each step.

Understanding the Equation


Before we begin solving the equation, let's take a closer look at its components. The equation consists of two fractions, each with a denominator that involves the sum or difference of two negative exponents. To simplify this equation, we need to first understand the properties of exponents and fractions.

Exponents and Fractions

Exponents are a shorthand way of representing repeated multiplication. For example, 232^3 is equivalent to 2Γ—2Γ—22 \times 2 \times 2. Negative exponents, on the other hand, represent the reciprocal of a number. In other words, 2βˆ’12^{-1} is equal to 12\frac{1}{2}.

Fractions, as we know, are a way of representing part of a whole. In this equation, we have two fractions, each with a denominator that involves the sum or difference of two negative exponents.

Simplifying the Denominators

To simplify the equation, we need to start by simplifying the denominators of each fraction. Let's begin with the first fraction:

2βˆ’12βˆ’1+3βˆ’1\frac{2^{-1}}{2^{-1} + 3^{-1}}

We can rewrite the denominator as follows:

2βˆ’1+3βˆ’1=12+132^{-1} + 3^{-1} = \frac{1}{2} + \frac{1}{3}

To add these fractions, we need to find a common denominator, which is 6. Therefore, we can rewrite the denominator as:

12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Now that we have simplified the denominator, we can rewrite the first fraction as:

2βˆ’12βˆ’1+3βˆ’1=2βˆ’156\frac{2^{-1}}{2^{-1} + 3^{-1}} = \frac{2^{-1}}{\frac{5}{6}}

Simplifying the First Fraction

To simplify the first fraction, we can multiply the numerator and denominator by 6, which is the reciprocal of the denominator:

2βˆ’156=2βˆ’1Γ—656Γ—6\frac{2^{-1}}{\frac{5}{6}} = \frac{2^{-1} \times 6}{\frac{5}{6} \times 6}

Simplifying the numerator and denominator, we get:

2βˆ’1Γ—656Γ—6=65\frac{2^{-1} \times 6}{\frac{5}{6} \times 6} = \frac{6}{5}

Simplifying the Second Fraction

Now that we have simplified the first fraction, let's move on to the second fraction:

2βˆ’12βˆ’1βˆ’3βˆ’1\frac{2^{-1}}{2^{-1} - 3^{-1}}

We can rewrite the denominator as follows:

2βˆ’1βˆ’3βˆ’1=12βˆ’132^{-1} - 3^{-1} = \frac{1}{2} - \frac{1}{3}

To subtract these fractions, we need to find a common denominator, which is 6. Therefore, we can rewrite the denominator as:

12βˆ’13=36βˆ’26=16\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}

Now that we have simplified the denominator, we can rewrite the second fraction as:

2βˆ’12βˆ’1βˆ’3βˆ’1=2βˆ’116\frac{2^{-1}}{2^{-1} - 3^{-1}} = \frac{2^{-1}}{\frac{1}{6}}

Simplifying the Second Fraction

To simplify the second fraction, we can multiply the numerator and denominator by 6, which is the reciprocal of the denominator:

2βˆ’116=2βˆ’1Γ—616Γ—6\frac{2^{-1}}{\frac{1}{6}} = \frac{2^{-1} \times 6}{\frac{1}{6} \times 6}

Simplifying the numerator and denominator, we get:

2βˆ’1Γ—616Γ—6=61=6\frac{2^{-1} \times 6}{\frac{1}{6} \times 6} = \frac{6}{1} = 6

Combining the Fractions


Now that we have simplified both fractions, we can combine them to get the final result:

2βˆ’12βˆ’1+3βˆ’1+2βˆ’12βˆ’1βˆ’3βˆ’1=65+6\frac{2^{-1}}{2^{-1} + 3^{-1}} + \frac{2^{-1}}{2^{-1} - 3^{-1}} = \frac{6}{5} + 6

To add these fractions, we need to find a common denominator, which is 5. Therefore, we can rewrite the second fraction as:

6=6Γ—55=3056 = \frac{6 \times 5}{5} = \frac{30}{5}

Now that we have a common denominator, we can add the fractions:

65+305=365\frac{6}{5} + \frac{30}{5} = \frac{36}{5}

Conclusion


In this article, we have solved the equation 2βˆ’12βˆ’1+3βˆ’1+2βˆ’12βˆ’1βˆ’3βˆ’1=185\frac{2^{-1}}{2^{-1} + 3^{-1}} + \frac{2^{-1}}{2^{-1} - 3^{-1}} = \frac{18}{5}. We started by simplifying the denominators of each fraction, then combined the fractions to get the final result. The final answer is 365\frac{36}{5}, which is equivalent to 7.2.

Final Answer

The final answer is 7.2\boxed{7.2}.

References

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Introduction


In our previous article, we solved the equation 2βˆ’12βˆ’1+3βˆ’1+2βˆ’12βˆ’1βˆ’3βˆ’1=185\frac{2^{-1}}{2^{-1} + 3^{-1}} + \frac{2^{-1}}{2^{-1} - 3^{-1}} = \frac{18}{5}. We simplified the denominators of each fraction, combined the fractions, and arrived at the final answer of 365\frac{36}{5}, which is equivalent to 7.2. In this article, we will answer some frequently asked questions about the equation and provide additional insights.

Q&A


Q: What is the meaning of the negative exponents in the equation?

A: Negative exponents represent the reciprocal of a number. In other words, 2βˆ’12^{-1} is equal to 12\frac{1}{2}.

Q: How do you simplify the denominators of the fractions?

A: To simplify the denominators, we need to find a common denominator, which is the least common multiple (LCM) of the two denominators. We can then rewrite the fractions with the common denominator and simplify.

Q: Can you provide an example of how to simplify the denominators?

A: Let's consider the first fraction:

2βˆ’12βˆ’1+3βˆ’1\frac{2^{-1}}{2^{-1} + 3^{-1}}

We can rewrite the denominator as follows:

2βˆ’1+3βˆ’1=12+132^{-1} + 3^{-1} = \frac{1}{2} + \frac{1}{3}

To add these fractions, we need to find a common denominator, which is 6. Therefore, we can rewrite the denominator as:

12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

Now that we have simplified the denominator, we can rewrite the first fraction as:

2βˆ’12βˆ’1+3βˆ’1=2βˆ’156\frac{2^{-1}}{2^{-1} + 3^{-1}} = \frac{2^{-1}}{\frac{5}{6}}

Q: How do you combine the fractions?

A: To combine the fractions, we need to find a common denominator, which is the least common multiple (LCM) of the two denominators. We can then rewrite the fractions with the common denominator and add them.

Q: Can you provide an example of how to combine the fractions?

A: Let's consider the two fractions:

2βˆ’12βˆ’1+3βˆ’1=65\frac{2^{-1}}{2^{-1} + 3^{-1}} = \frac{6}{5}

2βˆ’12βˆ’1βˆ’3βˆ’1=6\frac{2^{-1}}{2^{-1} - 3^{-1}} = 6

We can rewrite the second fraction as:

6=6Γ—55=3056 = \frac{6 \times 5}{5} = \frac{30}{5}

Now that we have a common denominator, we can add the fractions:

65+305=365\frac{6}{5} + \frac{30}{5} = \frac{36}{5}

Q: What is the final answer to the equation?

A: The final answer to the equation is 365\frac{36}{5}, which is equivalent to 7.2.

Q: Can you provide a summary of the steps to solve the equation?

A: Here is a summary of the steps to solve the equation:

  1. Simplify the denominators of each fraction.
  2. Combine the fractions by finding a common denominator.
  3. Add the fractions to get the final result.

Conclusion


In this article, we have answered some frequently asked questions about the equation and provided additional insights. We have also provided examples of how to simplify the denominators and combine the fractions. The final answer to the equation is 365\frac{36}{5}, which is equivalent to 7.2.

Final Answer

The final answer is 7.2\boxed{7.2}.

References

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