Simplify:${ \frac{\frac{\frac{1}{x+h}-\frac{1}{x}}{h}}{h} }$

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Introduction

In mathematics, complex fractions can be a daunting task to simplify. However, with a clear understanding of the concepts and a step-by-step approach, even the most complex fractions can be simplified. In this article, we will focus on simplifying the given complex fraction:

1x+h−1xhh\frac{\frac{\frac{1}{x+h}-\frac{1}{x}}{h}}{h}

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given complex fraction, we have a fraction in the numerator and a fraction in the denominator. To simplify this fraction, we need to follow a specific order of operations.

Order of Operations

The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the Complex Fraction

To simplify the given complex fraction, we need to follow the order of operations. Let's break it down step by step:

Step 1: Evaluate the Expressions Inside Parentheses

The first step is to evaluate the expressions inside the parentheses. In this case, we have two fractions inside the parentheses:

1x+h−1x\frac{1}{x+h}-\frac{1}{x}

To evaluate this expression, we need to find a common denominator. The common denominator is (x+h)x(x+h)x. Therefore, we can rewrite the expression as:

x−(x+h)(x+h)x\frac{x-(x+h)}{(x+h)x}

Simplifying the numerator, we get:

−h(x+h)x\frac{-h}{(x+h)x}

Step 2: Evaluate the Exponential Expressions

There are no exponential expressions in this complex fraction.

Step 3: Evaluate the Multiplication and Division Operations

The next step is to evaluate the multiplication and division operations. In this case, we have a fraction in the numerator and a fraction in the denominator. To simplify this fraction, we need to multiply the numerator and denominator by the reciprocal of the denominator.

−h(x+h)xh\frac{\frac{-h}{(x+h)x}}{h}

Multiplying the numerator and denominator by the reciprocal of the denominator, we get:

−hh(x+h)x\frac{-h}{h(x+h)x}

Step 4: Evaluate the Addition and Subtraction Operations

There are no addition and subtraction operations in this complex fraction.

Final Simplification

The final step is to simplify the fraction by canceling out any common factors. In this case, we can cancel out the hh in the numerator and denominator:

−1(x+h)x\frac{-1}{(x+h)x}

Conclusion

Simplifying complex fractions requires a clear understanding of the concepts and a step-by-step approach. By following the order of operations and simplifying the expressions inside parentheses, we can simplify even the most complex fractions. In this article, we simplified the given complex fraction:

1x+h−1xhh\frac{\frac{\frac{1}{x+h}-\frac{1}{x}}{h}}{h}

to:

−1(x+h)x\frac{-1}{(x+h)x}

Frequently Asked Questions

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, follow the order of operations: evaluate expressions inside parentheses, evaluate any exponential expressions, evaluate any multiplication and division operations, and finally, evaluate any addition and subtraction operations.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Additional Resources

For more information on simplifying complex fractions, check out the following resources:

  • Khan Academy: Simplifying Complex Fractions
  • Mathway: Simplifying Complex Fractions
  • Wolfram Alpha: Simplifying Complex Fractions

Final Thoughts

Simplifying complex fractions requires patience and practice. By following the order of operations and simplifying the expressions inside parentheses, we can simplify even the most complex fractions. Remember to always evaluate expressions inside parentheses first, followed by any exponential expressions, multiplication and division operations, and finally, addition and subtraction operations. With practice and persistence, you will become a master of simplifying complex fractions.

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Introduction

In our previous article, we explored the concept of simplifying complex fractions and provided a step-by-step guide on how to simplify the given complex fraction:

1x+h−1xhh\frac{\frac{\frac{1}{x+h}-\frac{1}{x}}{h}}{h}

However, we understand that simplifying complex fractions can be a challenging task, and many of you may have questions and doubts. In this article, we will address some of the most frequently asked questions about simplifying complex fractions.

Q&A

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, follow the order of operations: evaluate expressions inside parentheses, evaluate any exponential expressions, evaluate any multiplication and division operations, and finally, evaluate any addition and subtraction operations.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, follow the order of operations within the parentheses. For example, if you have an expression like:

1x+h−1x\frac{1}{x+h}-\frac{1}{x}

You would first find a common denominator, which is (x+h)x(x+h)x. Then, you would rewrite the expression as:

x−(x+h)(x+h)x\frac{x-(x+h)}{(x+h)x}

Simplifying the numerator, you get:

−h(x+h)x\frac{-h}{(x+h)x}

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, you can rewrite the fraction as:

1x−n=xn\frac{1}{x^{-n}} = x^n

For example, if you have a fraction like:

1x−2\frac{1}{x^{-2}}

You can rewrite it as:

x2x^2

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression a+ba+b is a−ba-b. For example, if you have a fraction like:

1x+2\frac{1}{x+\sqrt{2}}

You can multiply the numerator and denominator by the conjugate of the denominator, which is x−2x-\sqrt{2}:

1x+2⋅x−2x−2=x−2x2−2\frac{1}{x+\sqrt{2}} \cdot \frac{x-\sqrt{2}}{x-\sqrt{2}} = \frac{x-\sqrt{2}}{x^2-2}

Q: How do I simplify a complex fraction with multiple layers?

A: To simplify a complex fraction with multiple layers, you can follow the order of operations and simplify each layer one at a time. For example, if you have a complex fraction like:

1x+h−1xhh\frac{\frac{\frac{1}{x+h}-\frac{1}{x}}{h}}{h}

You can simplify the innermost layer first, which is:

1x+h−1xh\frac{\frac{1}{x+h}-\frac{1}{x}}{h}

Simplifying this layer, you get:

−hh(x+h)x\frac{-h}{h(x+h)x}

Then, you can simplify the next layer, which is:

−hh(x+h)x÷h\frac{-h}{h(x+h)x} \div h

Simplifying this layer, you get:

−1(x+h)x\frac{-1}{(x+h)x}

Conclusion

Simplifying complex fractions requires patience and practice. By following the order of operations and simplifying the expressions inside parentheses, you can simplify even the most complex fractions. We hope that this Q&A guide has helped you to better understand the concept of simplifying complex fractions and has provided you with the tools and confidence to tackle even the most challenging problems.

Additional Resources

For more information on simplifying complex fractions, check out the following resources:

  • Khan Academy: Simplifying Complex Fractions
  • Mathway: Simplifying Complex Fractions
  • Wolfram Alpha: Simplifying Complex Fractions

Final Thoughts

Simplifying complex fractions is a skill that takes time and practice to develop. However, with patience and persistence, you can master this skill and become a proficient mathematician. Remember to always follow the order of operations and simplify the expressions inside parentheses to simplify even the most complex fractions.