Simplify And Express With Positive Indices: ( A 2 3 B − 1 B 3 A − 2 ) ÷ ( A B − 4 B A − 2 ) 6 \left(\frac{a^{\frac{2}{3}} \sqrt{b^{-1}}}{b^3 \sqrt{a^{-2}}}\right) \div \left(\frac{a \sqrt{b^{-4}}}{b \sqrt{a^{-2}}}\right)^6 ( B 3 A − 2 ​ A 3 2 ​ B − 1 ​ ​ ) ÷ ( B A − 2 ​ A B − 4 ​ ​ ) 6

by ADMIN 287 views

=====================================================

Introduction

In mathematics, indices play a crucial role in simplifying complex expressions. When dealing with fractions and roots, it's essential to express them with positive indices to make calculations easier. In this article, we will simplify the given expression using positive indices and explore the concept of rational exponents.

Understanding Indices

Indices, also known as exponents, are a way to represent repeated multiplication of a number. For example, a3a^3 can be written as a×a×aa \times a \times a. When dealing with fractions and roots, indices can be used to simplify expressions. A positive index indicates that the number is being multiplied by itself, while a negative index indicates that the reciprocal of the number is being multiplied by itself.

Simplifying the Expression

The given expression is:

(a23b−1b3a−2)÷(ab−4ba−2)6\left(\frac{a^{\frac{2}{3}} \sqrt{b^{-1}}}{b^3 \sqrt{a^{-2}}}\right) \div \left(\frac{a \sqrt{b^{-4}}}{b \sqrt{a^{-2}}}\right)^6

To simplify this expression, we need to express the fractions and roots with positive indices.

Step 1: Simplify the Fractions

First, let's simplify the fractions in the expression.

a23b−1b3a−2=a23⋅b−12b3⋅a−2\frac{a^{\frac{2}{3}} \sqrt{b^{-1}}}{b^3 \sqrt{a^{-2}}} = \frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}}}{b^3 \cdot a^{-2}}

Step 2: Simplify the Roots

Next, let's simplify the roots in the expression.

b−1=b−12\sqrt{b^{-1}} = b^{-\frac{1}{2}}

a−2=a−1\sqrt{a^{-2}} = a^{-1}

b−4=b−2\sqrt{b^{-4}} = b^{-2}

a−2=a−1\sqrt{a^{-2}} = a^{-1}

Step 3: Substitute the Simplified Fractions and Roots

Now, let's substitute the simplified fractions and roots back into the original expression.

(a23⋅b−12b3⋅a−2)÷(a⋅b−2b⋅a−1)6\left(\frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}}}{b^3 \cdot a^{-2}}\right) \div \left(\frac{a \cdot b^{-2}}{b \cdot a^{-1}}\right)^6

Step 4: Simplify the Expression

To simplify the expression, we need to apply the rules of indices.

(a23⋅b−12b3⋅a−2)÷(a⋅b−2b⋅a−1)6=(a23⋅b−12b3⋅a−2)÷(a1⋅b−2b1⋅a−1)6\left(\frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}}}{b^3 \cdot a^{-2}}\right) \div \left(\frac{a \cdot b^{-2}}{b \cdot a^{-1}}\right)^6 = \left(\frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}}}{b^3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b^{-2}}{b^1 \cdot a^{-1}}\right)^6

Step 5: Apply the Rules of Indices

Now, let's apply the rules of indices to simplify the expression.

$\left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b{-\frac{1}{2}}}{b3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b{-2}}{b1 \cdot a{-1}}\right)6 = \left(\frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}}

=====================================================

Introduction

In our previous article, we simplified the given expression using positive indices and explored the concept of rational exponents. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What are positive indices?

A: Positive indices are a way to represent repeated multiplication of a number. For example, a3a^3 can be written as a×a×aa \times a \times a. When dealing with fractions and roots, positive indices can be used to simplify expressions.

Q: How do I simplify fractions with indices?

A: To simplify fractions with indices, you need to apply the rules of indices. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: What is the difference between a positive index and a negative index?

A: A positive index indicates that the number is being multiplied by itself, while a negative index indicates that the reciprocal of the number is being multiplied by itself.

Q: How do I simplify roots with indices?

A: To simplify roots with indices, you need to apply the rules of indices. For example, am=am2\sqrt{a^m} = a^{\frac{m}{2}}.

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

A: When dividing two numbers with indices, you need to subtract the indices. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

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

A: When multiplying two numbers with indices, you need to add the indices. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: How do I simplify an expression with multiple fractions and roots?

A: To simplify an expression with multiple fractions and roots, you need to apply the rules of indices and simplify each fraction and root separately.

Q: What is the importance of expressing fractions and roots with positive indices?

A: Expressing fractions and roots with positive indices makes it easier to simplify complex expressions and perform calculations.

Q: Can you provide an example of simplifying an expression with positive indices?

A: Let's consider the expression (a23⋅b−12b3⋅a−2)÷(a1⋅b−2b1⋅a−1)6\left(\frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}}}{b^3 \cdot a^{-2}}\right) \div \left(\frac{a^1 \cdot b^{-2}}{b^1 \cdot a^{-1}}\right)^6. We can simplify this expression by applying the rules of indices and simplifying each fraction and root separately.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with positive indices. We hope that this article has provided you with a better understanding of the concept of rational exponents and how to simplify complex expressions.

Additional Resources

Final Thoughts

Simplifying expressions with positive indices is an essential skill in mathematics. By understanding the rules of indices and how to simplify fractions and roots, you can perform complex calculations with ease. We hope that this article has provided you with a solid foundation in this topic and has inspired you to learn more about mathematics.