Which Shows The Following Expression After The Negative Exponents Have Been Eliminated?$\[ \frac{a^3 B^{-2}}{a B^{-4}}, \quad A \neq 0, \, B \neq 0 \\]A. \[$\frac{a^3 B^4}{a B^2}\$\]B. \[$\frac{a^3 B^{-4}}{a B^{-2}}\$\]C.

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Understanding Negative Exponents

In mathematics, negative exponents are a way to represent very small numbers or fractions. When we have a negative exponent, it means that the base number is being raised to a power that is the opposite of the given exponent. For example, a−2a^{-2} is equal to 1a2\frac{1}{a^2}. In this article, we will explore how to simplify expressions with negative exponents.

The Given Expression

The given expression is a3b−2ab−4\frac{a^3 b^{-2}}{a b^{-4}}, where a≠0a \neq 0 and b≠0b \neq 0. Our goal is to simplify this expression by eliminating the negative exponents.

Step 1: Simplify the Numerator

To simplify the numerator, we can use the rule of exponents that states amâ‹…an=am+na^m \cdot a^n = a^{m+n}. Applying this rule to the numerator, we get:

a3b−2=a3⋅b−2=a3−2b−2=a1b−2=ab2a^3 b^{-2} = a^3 \cdot b^{-2} = a^{3-2} b^{-2} = a^1 b^{-2} = \frac{a}{b^2}

Step 2: Simplify the Denominator

To simplify the denominator, we can use the rule of exponents that states aman=am−n\frac{a^m}{a^n} = a^{m-n}. Applying this rule to the denominator, we get:

ab−4=a⋅b−4=a1−4b−4=a−3b−4=1a3b4a b^{-4} = a \cdot b^{-4} = a^{1-4} b^{-4} = a^{-3} b^{-4} = \frac{1}{a^3 b^4}

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can rewrite the original expression as:

a3b−2ab−4=ab21a3b4\frac{a^3 b^{-2}}{a b^{-4}} = \frac{\frac{a}{b^2}}{\frac{1}{a^3 b^4}}

To simplify this expression further, we can multiply the numerator and denominator by a3b4a^3 b^4:

ab21a3b4=ab2â‹…a3b41=a4b4b2=a4b2\frac{\frac{a}{b^2}}{\frac{1}{a^3 b^4}} = \frac{a}{b^2} \cdot \frac{a^3 b^4}{1} = \frac{a^4 b^4}{b^2} = a^4 b^2

Conclusion

In conclusion, the expression a3b−2ab−4\frac{a^3 b^{-2}}{a b^{-4}} simplifies to a4b2a^4 b^2 after eliminating the negative exponents.

Answer

The correct answer is:

  • A. a3b4ab2\frac{a^3 b^4}{a b^2}

This answer is incorrect because it does not eliminate the negative exponents.

  • B. a3b−4ab−2\frac{a^3 b^{-4}}{a b^{-2}}

This answer is incorrect because it does not eliminate the negative exponents.

  • C. a4b2a^4 b^2

This answer is correct because it eliminates the negative exponents and simplifies the expression.

Discussion

The concept of negative exponents can be challenging to understand, especially for students who are new to algebra. However, with practice and patience, anyone can master the rules of exponents and simplify expressions with negative exponents.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions with negative exponents:

  • Use the rule of exponents that states amâ‹…an=am+na^m \cdot a^n = a^{m+n} to simplify the numerator.
  • Use the rule of exponents that states aman=am−n\frac{a^m}{a^n} = a^{m-n} to simplify the denominator.
  • Multiply the numerator and denominator by the same expression to eliminate the negative exponents.
  • Simplify the expression by canceling out any common factors.

Q: What is a negative exponent?

A: A negative exponent is a way to represent very small numbers or fractions. When we have a negative exponent, it means that the base number is being raised to a power that is the opposite of the given exponent.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the rule of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. You can also use the rule of exponents that states aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: What is the rule for multiplying two numbers with negative exponents?

A: When multiplying two numbers with negative exponents, you can add the exponents. For example, a−2⋅a−3=a−2−3=a−5a^{-2} \cdot a^{-3} = a^{-2-3} = a^{-5}.

Q: What is the rule for dividing two numbers with negative exponents?

A: When dividing two numbers with negative exponents, you can subtract the exponents. For example, a−2a−3=a−2−(−3)=a−2+3=a1\frac{a^{-2}}{a^{-3}} = a^{-2-(-3)} = a^{-2+3} = a^{1}.

Q: How do I simplify an expression with a negative exponent in the denominator?

A: To simplify an expression with a negative exponent in the denominator, you can multiply the numerator and denominator by the same expression to eliminate the negative exponent.

Q: What is the difference between a negative exponent and a fraction?

A: A negative exponent is a way to represent a very small number or fraction, while a fraction is a way to represent a part of a whole. For example, a−2a^{-2} is equal to 1a2\frac{1}{a^2}.

Q: Can I simplify an expression with a negative exponent by canceling out common factors?

A: Yes, you can simplify an expression with a negative exponent by canceling out common factors. For example, a3b−2ab−4=a3a⋅b−2b−4=a2⋅b2=a2b2\frac{a^3 b^{-2}}{a b^{-4}} = \frac{a^3}{a} \cdot \frac{b^{-2}}{b^{-4}} = a^2 \cdot b^2 = a^2 b^2.

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

A: Some common mistakes to avoid when simplifying expressions with negative exponents include:

  • Not using the rule of exponents to simplify the expression
  • Not multiplying the numerator and denominator by the same expression to eliminate the negative exponent
  • Not canceling out common factors
  • Not using the correct rules for multiplying and dividing numbers with negative exponents

Q: How can I practice simplifying expressions with negative exponents?

A: You can practice simplifying expressions with negative exponents by working through examples and exercises in your textbook or online resources. You can also try simplifying expressions with negative exponents on your own by using the rules and techniques discussed in this article.

Q: What are some real-world applications of simplifying expressions with negative exponents?

A: Simplifying expressions with negative exponents has many real-world applications, including:

  • Calculating interest rates and investments
  • Determining the area and volume of shapes
  • Solving problems in physics and engineering
  • Working with financial data and statistics

By understanding and applying the rules for simplifying expressions with negative exponents, you can solve a wide range of problems and make informed decisions in your personal and professional life.