Simplify The Expression:$\left(\frac{a^3 B^{-6}}{a^{-3} B^3}\right)^{\frac{1}{3}}$Write Your Answer Without Negative Exponents.

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Introduction

In this article, we will simplify the given expression (a3b−6a−3b3)13\left(\frac{a^3 b^{-6}}{a^{-3} b^3}\right)^{\frac{1}{3}} and provide the final answer without negative exponents. This involves applying the rules of exponents and simplifying the expression step by step.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, a3a^3 means a×a×aa \times a \times a. When we have a fraction with exponents, we can simplify it by applying the rules of exponents.

Simplifying the Expression

To simplify the given expression, we will start by applying the quotient rule of exponents, which states that when we divide two powers with the same base, we subtract the exponents.

(a3b−6a−3b3)13\left(\frac{a^3 b^{-6}}{a^{-3} b^3}\right)^{\frac{1}{3}}

Using the quotient rule, we can rewrite the expression as:

(a3−(−3)b−6−3)13\left(a^{3-(-3)} b^{-6-3}\right)^{\frac{1}{3}}

Simplifying the exponents, we get:

(a6b−9)13\left(a^6 b^{-9}\right)^{\frac{1}{3}}

Applying the Power Rule

The power rule of exponents states that when we raise a power to another power, we multiply the exponents. In this case, we have a power raised to the power of 13\frac{1}{3}.

(a6b−9)13\left(a^6 b^{-9}\right)^{\frac{1}{3}}

Using the power rule, we can rewrite the expression as:

a6×13b−9×13a^{6 \times \frac{1}{3}} b^{-9 \times \frac{1}{3}}

Simplifying the exponents, we get:

a2b−3a^2 b^{-3}

Removing Negative Exponents

To remove the negative exponent, we can rewrite the expression as:

a2b3\frac{a^2}{b^3}

This is the final simplified expression without negative exponents.

Conclusion

In this article, we simplified the given expression (a3b−6a−3b3)13\left(\frac{a^3 b^{-6}}{a^{-3} b^3}\right)^{\frac{1}{3}} and provided the final answer without negative exponents. We applied the rules of exponents, including the quotient rule and the power rule, to simplify the expression step by step.

Frequently Asked Questions

Q: What is the simplified expression without negative exponents?

A: The simplified expression without negative exponents is a2b3\frac{a^2}{b^3}.

Q: How do I apply the quotient rule of exponents?

A: To apply the quotient rule of exponents, you subtract the exponents when dividing two powers with the same base.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, you multiply the exponents when raising a power to another power.

Final Answer

The final answer is a2b3\boxed{\frac{a^2}{b^3}}.

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Introduction

In our previous article, we simplified the given expression (a3b−6a−3b3)13\left(\frac{a^3 b^{-6}}{a^{-3} b^3}\right)^{\frac{1}{3}} and provided the final answer without negative exponents. In this article, we will provide a Q&A guide to help you understand the concepts and rules of exponents used in simplifying the expression.

Q&A Guide

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that when we divide two powers with the same base, we subtract the exponents. For example, a3a2=a3−2=a1\frac{a^3}{a^2} = a^{3-2} = a^1.

Q: How do I apply the quotient rule of exponents?

A: To apply the quotient rule of exponents, you subtract the exponents when dividing two powers with the same base. For example, a3b−6a−3b3=a3−(−3)b−6−3=a6b−9\frac{a^3 b^{-6}}{a^{-3} b^3} = a^{3-(-3)} b^{-6-3} = a^6 b^{-9}.

Q: What is the power rule of exponents?

A: The power rule of exponents states that when we raise a power to another power, we multiply the exponents. For example, (a2)3=a2×3=a6(a^2)^3 = a^{2 \times 3} = a^6.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, you multiply the exponents when raising a power to another power. For example, (a6b−9)13=a6×13b−9×13=a2b−3(a^6 b^{-9})^{\frac{1}{3}} = a^{6 \times \frac{1}{3}} b^{-9 \times \frac{1}{3}} = a^2 b^{-3}.

Q: How do I remove negative exponents?

A: To remove negative exponents, you can rewrite the expression as a fraction with a positive exponent. For example, a2b−3=a2b3a^2 b^{-3} = \frac{a^2}{b^3}.

Q: What is the final simplified expression without negative exponents?

A: The final simplified expression without negative exponents is a2b3\frac{a^2}{b^3}.

Common Mistakes

Mistake 1: Not applying the quotient rule of exponents

  • When dividing two powers with the same base, make sure to subtract the exponents.
  • Example: a3a2=a3−2=a1\frac{a^3}{a^2} = a^{3-2} = a^1, not a3a^3.

Mistake 2: Not applying the power rule of exponents

  • When raising a power to another power, make sure to multiply the exponents.
  • Example: (a2)3=a2×3=a6(a^2)^3 = a^{2 \times 3} = a^6, not a2a^2.

Mistake 3: Not removing negative exponents

  • When removing negative exponents, make sure to rewrite the expression as a fraction with a positive exponent.
  • Example: a2b−3=a2b3a^2 b^{-3} = \frac{a^2}{b^3}, not a2b3a^2 b^3.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and rules of exponents used in simplifying the expression (a3b−6a−3b3)13\left(\frac{a^3 b^{-6}}{a^{-3} b^3}\right)^{\frac{1}{3}}. We also discussed common mistakes to avoid when working with exponents.

Final Answer

The final answer is a2b3\boxed{\frac{a^2}{b^3}}.