Simplify. $\frac{b^2}{3 A^{-4}}$A. $-\frac{3 B}{a^4}$ B. $\frac{a^4 B^2}{3}$ C. $\frac{3}{(a B)^8}$ D. $\frac{3}{a B^8}$

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Introduction

Simplifying complex fractions can be a daunting task, especially when dealing with exponents and negative numbers. However, with a clear understanding of the rules and a step-by-step approach, you can simplify even the most complex fractions with ease. In this article, we will explore the process of simplifying complex fractions, using the example of b23a−4\frac{b^2}{3 a^{-4}}.

Understanding Exponents

Before we dive into simplifying the complex fraction, it's essential to understand the rules of exponents. An exponent is a small number that is placed above and to the right of a number or a variable, indicating how many times the base is multiplied by itself. For example, a4a^4 means aa multiplied by itself four times: a×a×a×aa \times a \times a \times a.

Simplifying the Complex Fraction

Now that we have a basic understanding of exponents, let's simplify the complex fraction b23a−4\frac{b^2}{3 a^{-4}}. To simplify this fraction, we need to follow the order of operations (PEMDAS):

  1. Parentheses: There are no parentheses in this expression, so we can move on to the next step.
  2. Exponents: We need to simplify the exponent a−4a^{-4}. According to the rules of exponents, a−4a^{-4} means 1a4\frac{1}{a^4}.
  3. Multiplication and Division: We can now rewrite the fraction as b23×1a4\frac{b^2}{3} \times \frac{1}{a^4}.
  4. Simplification: We can simplify the fraction by multiplying the numerators and denominators: b2×13×a4=b23a4\frac{b^2 \times 1}{3 \times a^4} = \frac{b^2}{3 a^4}.

Answer

So, what is the simplified form of b23a−4\frac{b^2}{3 a^{-4}}? The answer is b23a4\frac{b^2}{3 a^4}.

Comparison with Answer Choices

Now that we have simplified the complex fraction, let's compare our answer with the answer choices:

  • A. −3ba4-\frac{3 b}{a^4}: This is not the correct answer, as we did not introduce any negative signs during the simplification process.
  • B. a4b23\frac{a^4 b^2}{3}: This is not the correct answer, as we did not swap the numerator and denominator during the simplification process.
  • C. 3(ab)8\frac{3}{(a b)^8}: This is not the correct answer, as we did not introduce any additional exponents or parentheses during the simplification process.
  • D. 3ab8\frac{3}{a b^8}: This is not the correct answer, as we did not introduce any additional exponents or parentheses during the simplification process.

Conclusion

Simplifying complex fractions requires a clear understanding of the rules of exponents and a step-by-step approach. By following the order of operations (PEMDAS) and simplifying the exponent a−4a^{-4}, we arrived at the simplified form of b23a4\frac{b^2}{3 a^4}. This example demonstrates the importance of careful attention to detail and a thorough understanding of mathematical concepts.

Additional Examples

To further reinforce your understanding of simplifying complex fractions, let's consider a few additional examples:

  • a32b−2\frac{a^3}{2 b^{-2}}: Simplify this complex fraction using the rules of exponents.
  • c24d−1\frac{c^2}{4 d^{-1}}: Simplify this complex fraction using the rules of exponents.
  • e43f−3\frac{e^4}{3 f^{-3}}: Simplify this complex fraction using the rules of exponents.

Practice Problems

To practice simplifying complex fractions, try the following problems:

  • Simplify g25h−4\frac{g^2}{5 h^{-4}}.
  • Simplify i32j−2\frac{i^3}{2 j^{-2}}.
  • Simplify k43l−1\frac{k^4}{3 l^{-1}}.

Conclusion

Introduction

Simplifying complex fractions can be a challenging task, especially when dealing with exponents and negative numbers. However, with a clear understanding of the rules and a step-by-step approach, you can simplify even the most complex fractions with ease. In this article, we will answer some frequently asked questions about simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. For example, b23a−4\frac{b^2}{3 a^{-4}} is a complex fraction because it contains a fraction in its numerator and a fraction in its denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses.
  2. Exponents: Simplify any exponents in the numerator and denominator.
  3. Multiplication and Division: Perform any multiplication and division operations from left to right.
  4. Addition and Subtraction: Perform any addition and subtraction operations from left to right.

Q: What is the rule for simplifying exponents?

A: The rule for simplifying exponents is as follows:

  • am×an=am+na^m \times a^n = a^{m+n}
  • aman=am−n\frac{a^m}{a^n} = a^{m-n}
  • a−m=1ama^{-m} = \frac{1}{a^m}

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

A: To simplify a fraction with a negative exponent, you need to follow the rule a−m=1ama^{-m} = \frac{1}{a^m}. For example, 1a−4=a4\frac{1}{a^{-4}} = a^4.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors. For example, 2a24a2=12\frac{2a^2}{4a^2} = \frac{1}{2}.

Q: What is the difference between simplifying a complex fraction and canceling out common factors?

A: Simplifying a complex fraction involves following the order of operations (PEMDAS) and simplifying exponents, while canceling out common factors involves canceling out common factors in the numerator and denominator.

Q: Can I simplify a complex fraction with a variable in the denominator?

A: Yes, you can simplify a complex fraction with a variable in the denominator. For example, b23a−4\frac{b^2}{3a^{-4}} can be simplified by following the order of operations (PEMDAS) and simplifying exponents.

Q: How do I know when to simplify a complex fraction?

A: You should simplify a complex fraction when:

  • The fraction is difficult to work with in its current form.
  • The fraction contains a variable in the denominator.
  • The fraction contains a negative exponent.

Conclusion

Simplifying complex fractions is a crucial skill in mathematics, and with practice and patience, you can master this technique. By following the order of operations (PEMDAS) and simplifying exponents, you can simplify even the most complex fractions with ease. Remember to always double-check your work and compare your answer with the answer choices to ensure accuracy.

Practice Problems

To practice simplifying complex fractions, try the following problems:

  • Simplify g25h−4\frac{g^2}{5h^{-4}}.
  • Simplify i32j−2\frac{i^3}{2j^{-2}}.
  • Simplify k43l−1\frac{k^4}{3l^{-1}}.

Additional Resources

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

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