Simplify The Expression: ( 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

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Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified to reveal their underlying structure. In this article, we will focus on simplifying a specific expression involving exponents, radicals, and fractions. We will break down the expression into manageable parts, apply the rules of exponents and radicals, and ultimately arrive at a simplified form.

Understanding 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

This expression involves several components, including exponents, radicals, and fractions. To simplify it, we need to understand the properties of exponents and radicals, as well as the rules for dividing fractions.

Properties of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, a3a^3 can be written as a⋅a⋅aa \cdot a \cdot a. When we have multiple exponents with the same base, we can add or subtract them by multiplying or dividing the exponents, respectively. For example:

a3⋅a2=a3+2=a5a^3 \cdot a^2 = a^{3+2} = a^5

a3÷a2=a3−2=a1a^3 \div a^2 = a^{3-2} = a^1

Properties of Radicals

Radicals, such as square roots and cube roots, are a way of representing the inverse operation of exponentiation. For example, a\sqrt{a} is equivalent to a12a^{\frac{1}{2}}. When we have multiple radicals with the same base, we can multiply or divide them by multiplying or dividing the exponents, respectively. For example:

a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}

a÷b=ab\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}

Simplifying the Expression

Now that we have a good understanding of the properties of exponents and radicals, we can begin to simplify the given expression.

Step 1: Simplify the First Fraction

The first fraction is:

a23b−1b3a−2\frac{a^{\frac{2}{3}} \sqrt{b^{-1}}}{b^3 \sqrt{a^{-2}}}

We can simplify this fraction by applying the rules of exponents and radicals. First, we can rewrite the numerator and denominator using the properties of exponents:

a23b−1b3a−2=a23⋅b−12b3⋅a−1\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^{-1}}

Next, we can apply the rule for dividing fractions by multiplying the numerator and denominator by the reciprocal of the denominator:

a23⋅b−12b3⋅a−1=a23⋅b−12⋅ab3⋅a−1⋅a\frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}}}{b^3 \cdot a^{-1}} = \frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}} \cdot a}{b^3 \cdot a^{-1} \cdot a}

Simplifying further, we get:

a23⋅b−12⋅ab3⋅a−1⋅a=a53⋅b−12b3\frac{a^{\frac{2}{3}} \cdot b^{-\frac{1}{2}} \cdot a}{b^3 \cdot a^{-1} \cdot a} = \frac{a^{\frac{5}{3}} \cdot b^{-\frac{1}{2}}}{b^3}

Step 2: Simplify the Second Fraction

The second fraction is:

(ab−4ba−2)6\left(\frac{a \sqrt{b^{-4}}}{b \sqrt{a^{-2}}}\right)^6

We can simplify this fraction by applying the rules of exponents and radicals. First, we can rewrite the numerator and denominator using the properties of exponents:

(ab−4ba−2)6=(a⋅b−2b⋅a−1)6\left(\frac{a \sqrt{b^{-4}}}{b \sqrt{a^{-2}}}\right)^6 = \left(\frac{a \cdot b^{-2}}{b \cdot a^{-1}}\right)^6

Next, we can apply the rule for raising a fraction to a power by raising the numerator and denominator to that power:

(a⋅b−2b⋅a−1)6=(a⋅b−2)6(b⋅a−1)6\left(\frac{a \cdot b^{-2}}{b \cdot a^{-1}}\right)^6 = \frac{(a \cdot b^{-2})^6}{(b \cdot a^{-1})^6}

Simplifying further, we get:

(a⋅b−2)6(b⋅a−1)6=a6⋅b−12b6⋅a−6\frac{(a \cdot b^{-2})^6}{(b \cdot a^{-1})^6} = \frac{a^6 \cdot b^{-12}}{b^6 \cdot a^{-6}}

Step 3: Simplify the Expression

Now that we have simplified the first and second fractions, we can combine them to simplify the original expression.

(a53⋅b−12b3)÷(a6⋅b−12b6⋅a−6)\left(\frac{a^{\frac{5}{3}} \cdot b^{-\frac{1}{2}}}{b^3}\right) \div \left(\frac{a^6 \cdot b^{-12}}{b^6 \cdot a^{-6}}\right)

We can simplify this expression by applying the rule for dividing fractions by multiplying the numerator and denominator by the reciprocal of the denominator:

(a53⋅b−12b3)÷(a6⋅b−12b6⋅a−6)=(a53⋅b−12b3)⋅(b6⋅a−6a6⋅b−12)\left(\frac{a^{\frac{5}{3}} \cdot b^{-\frac{1}{2}}}{b^3}\right) \div \left(\frac{a^6 \cdot b^{-12}}{b^6 \cdot a^{-6}}\right) = \left(\frac{a^{\frac{5}{3}} \cdot b^{-\frac{1}{2}}}{b^3}\right) \cdot \left(\frac{b^6 \cdot a^{-6}}{a^6 \cdot b^{-12}}\right)

Simplifying further, we get:

(a53⋅b−12b3)⋅(b6⋅a−6a6⋅b−12)=a53⋅b−12⋅b6⋅a−6b3⋅a6⋅b−12\left(\frac{a^{\frac{5}{3}} \cdot b^{-\frac{1}{2}}}{b^3}\right) \cdot \left(\frac{b^6 \cdot a^{-6}}{a^6 \cdot b^{-12}}\right) = \frac{a^{\frac{5}{3}} \cdot b^{-\frac{1}{2}} \cdot b^6 \cdot a^{-6}}{b^3 \cdot a^6 \cdot b^{-12}}

Step 4: Final Simplification

We can simplify this expression further by applying the rule for multiplying fractions by multiplying the numerators and denominators:

a53⋅b−12⋅b6⋅a−6b3⋅a6⋅b−12=a53−6⋅b6−(−12)−31\frac{a^{\frac{5}{3}} \cdot b^{-\frac{1}{2}} \cdot b^6 \cdot a^{-6}}{b^3 \cdot a^6 \cdot b^{-12}} = \frac{a^{\frac{5}{3}-6} \cdot b^{6-(-\frac{1}{2})-3}}{1}

Simplifying further, we get:

a53−6⋅b6−(−12)−31=a−113⋅b132\frac{a^{\frac{5}{3}-6} \cdot b^{6-(-\frac{1}{2})-3}}{1} = a^{-\frac{11}{3}} \cdot b^{\frac{13}{2}}

Conclusion

In this article, we simplified a complex algebraic expression involving exponents, radicals, and fractions. We applied the rules of exponents and radicals, as well as the rules for dividing fractions, to arrive at a simplified form. The final simplified expression is:

a−113⋅b132a^{-\frac{11}{3}} \cdot b^{\frac{13}{2}}

This expression can be further simplified by applying the rules of exponents and radicals, but for the purposes of this article, we have arrived at a simplified form that reveals the underlying structure of the original expression.

Introduction

In our previous article, we simplified a complex algebraic expression involving exponents, radicals, and fractions. We applied the rules of exponents and radicals, as well as the rules for dividing fractions, to arrive at a simplified form. In this article, we will answer some common questions that readers may have about the simplification process.

Q&A

Q: What is the difference between an exponent and a radical?

A: An exponent is a shorthand way of representing repeated multiplication. For example, a3a^3 can be written as a⋅a⋅aa \cdot a \cdot a. A radical, on the other hand, is a way of representing the inverse operation of exponentiation. For example, a\sqrt{a} is equivalent to a12a^{\frac{1}{2}}.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can add or subtract the exponents by multiplying or dividing them, respectively. For example:

a3⋅a2=a3+2=a5a^3 \cdot a^2 = a^{3+2} = a^5

a3÷a2=a3−2=a1a^3 \div a^2 = a^{3-2} = a^1

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you can multiply or divide the radicals by multiplying or dividing the exponents, respectively. For example:

a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}

a÷b=ab\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}

Q: What is the rule for dividing fractions?

A: The rule for dividing fractions is to multiply the numerator and denominator by the reciprocal of the denominator. For example:

ab÷cd=ab⋅dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Q: How do I simplify an expression with a fraction raised to a power?

A: To simplify an expression with a fraction raised to a power, you can raise the numerator and denominator to that power. For example:

(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Q: What is the final simplified expression?

A: The final simplified expression is:

a−113⋅b132a^{-\frac{11}{3}} \cdot b^{\frac{13}{2}}

This expression can be further simplified by applying the rules of exponents and radicals, but for the purposes of this article, we have arrived at a simplified form that reveals the underlying structure of the original expression.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
  • Not simplifying radicals: Make sure to simplify radicals by multiplying or dividing the exponents.
  • Not applying the rules of exponents: Make sure to apply the rules of exponents when simplifying expressions with multiple exponents.
  • Not using the correct notation: Make sure to use the correct notation when writing expressions, including parentheses and exponents.

Conclusion

In this article, we answered some common questions that readers may have about the simplification process. We also highlighted some common mistakes to avoid when simplifying algebraic expressions. By following the rules of exponents and radicals, as well as the rules for dividing fractions, you can simplify complex algebraic expressions and arrive at a simplified form.