Simplify The Expression:${ \left(\frac{2 X^{-3} Y 2}{3 {-1} Y 3}\right) 2 \left(\frac{4 X^{-2} Y^3}{3 X 5}\right) 3 \div \left(\frac{81 X {-2}}{y {-3}}\right)^{-2} }$

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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 delve into the world of algebraic manipulation and explore the steps involved in simplifying a given expression. We will use the expression (2x−3y23−1y3)2(4x−2y33x5)3÷(81x−2y−3)−2\left(\frac{2 x^{-3} y^2}{3^{-1} y^3}\right)^2 \left(\frac{4 x^{-2} y^3}{3 x^5}\right)^3 \div \left(\frac{81 x^{-2}}{y^{-3}}\right)^{-2} as a case study and demonstrate how to simplify it using various algebraic techniques.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The expression consists of three main parts:

  1. (2x−3y23−1y3)2\left(\frac{2 x^{-3} y^2}{3^{-1} y^3}\right)^2
  2. (4x−2y33x5)3\left(\frac{4 x^{-2} y^3}{3 x^5}\right)^3
  3. (81x−2y−3)−2\left(\frac{81 x^{-2}}{y^{-3}}\right)^{-2}

Each of these parts involves exponents, fractions, and variables, making it a challenging expression to simplify.

Simplifying the First Part

Let's start by simplifying the first part of the expression: (2x−3y23−1y3)2\left(\frac{2 x^{-3} y^2}{3^{-1} y^3}\right)^2. To simplify this expression, we need to apply the exponent rule for fractions, which states that (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.

Using this rule, we can rewrite the first part of the expression as:

(2x−3y23−1y3)2=(2x−3y2)2(3−1y3)2\left(\frac{2 x^{-3} y^2}{3^{-1} y^3}\right)^2 = \frac{(2 x^{-3} y^2)^2}{(3^{-1} y^3)^2}

Now, let's simplify the numerator and denominator separately. The numerator can be simplified using the exponent rule for products, which states that (ab)n=anbn(ab)^n = a^n b^n. Applying this rule, we get:

(2x−3y2)2=22(x−3)2(y2)2=4x−6y4(2 x^{-3} y^2)^2 = 2^2 (x^{-3})^2 (y^2)^2 = 4 x^{-6} y^4

The denominator can be simplified using the exponent rule for powers of powers, which states that (am)n=amn(a^m)^n = a^{mn}. Applying this rule, we get:

(3−1y3)2=(3−1)2(y3)2=3−2y6(3^{-1} y^3)^2 = (3^{-1})^2 (y^3)^2 = 3^{-2} y^6

Now, let's substitute these simplified expressions back into the first part of the expression:

(2x−3y2)2(3−1y3)2=4x−6y43−2y6\frac{(2 x^{-3} y^2)^2}{(3^{-1} y^3)^2} = \frac{4 x^{-6} y^4}{3^{-2} y^6}

Simplifying the Second Part

Next, let's simplify the second part of the expression: (4x−2y33x5)3\left(\frac{4 x^{-2} y^3}{3 x^5}\right)^3. To simplify this expression, we need to apply the exponent rule for fractions, which states that (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.

Using this rule, we can rewrite the second part of the expression as:

(4x−2y33x5)3=(4x−2y3)3(3x5)3\left(\frac{4 x^{-2} y^3}{3 x^5}\right)^3 = \frac{(4 x^{-2} y^3)^3}{(3 x^5)^3}

Now, let's simplify the numerator and denominator separately. The numerator can be simplified using the exponent rule for products, which states that (ab)n=anbn(ab)^n = a^n b^n. Applying this rule, we get:

(4x−2y3)3=43(x−2)3(y3)3=64x−6y9(4 x^{-2} y^3)^3 = 4^3 (x^{-2})^3 (y^3)^3 = 64 x^{-6} y^9

The denominator can be simplified using the exponent rule for powers of powers, which states that (am)n=amn(a^m)^n = a^{mn}. Applying this rule, we get:

(3x5)3=33(x5)3=27x15(3 x^5)^3 = 3^3 (x^5)^3 = 27 x^{15}

Now, let's substitute these simplified expressions back into the second part of the expression:

(4x−2y3)3(3x5)3=64x−6y927x15\frac{(4 x^{-2} y^3)^3}{(3 x^5)^3} = \frac{64 x^{-6} y^9}{27 x^{15}}

Simplifying the Third Part

Finally, let's simplify the third part of the expression: (81x−2y−3)−2\left(\frac{81 x^{-2}}{y^{-3}}\right)^{-2}. To simplify this expression, we need to apply the exponent rule for fractions, which states that (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.

Using this rule, we can rewrite the third part of the expression as:

(81x−2y−3)−2=(81x−2)−2(y−3)−2\left(\frac{81 x^{-2}}{y^{-3}}\right)^{-2} = \frac{(81 x^{-2})^{-2}}{(y^{-3})^{-2}}

Now, let's simplify the numerator and denominator separately. The numerator can be simplified using the exponent rule for powers of powers, which states that (am)n=amn(a^m)^n = a^{mn}. Applying this rule, we get:

(81x−2)−2=(81)−2(x−2)−2=1812x4(81 x^{-2})^{-2} = (81)^{-2} (x^{-2})^{-2} = \frac{1}{81^2} x^4

The denominator can be simplified using the exponent rule for powers of powers, which states that (am)n=amn(a^m)^n = a^{mn}. Applying this rule, we get:

(y−3)−2=(y−3)2=y6(y^{-3})^{-2} = (y^{-3})^2 = y^6

Now, let's substitute these simplified expressions back into the third part of the expression:

(81x−2)−2(y−3)−2=1812x4y6\frac{(81 x^{-2})^{-2}}{(y^{-3})^{-2}} = \frac{\frac{1}{81^2} x^4}{y^6}

Combining the Simplified Expressions

Now that we have simplified each part of the expression, let's combine them to get the final simplified expression.

(2x−3y23−1y3)2(4x−2y33x5)3÷(81x−2y−3)−2=4x−6y43−2y6⋅64x−6y927x15⋅y61812x4\left(\frac{2 x^{-3} y^2}{3^{-1} y^3}\right)^2 \left(\frac{4 x^{-2} y^3}{3 x^5}\right)^3 \div \left(\frac{81 x^{-2}}{y^{-3}}\right)^{-2} = \frac{4 x^{-6} y^4}{3^{-2} y^6} \cdot \frac{64 x^{-6} y^9}{27 x^{15}} \cdot \frac{y^6}{\frac{1}{81^2} x^4}

Final Simplification

To simplify this expression further, we need to combine like terms and apply the exponent rules.

4x−6y43−2y6⋅64x−6y927x15⋅y61812x4=4⋅64⋅y4+9+63−2⋅27⋅x−6−6+4⋅1812\frac{4 x^{-6} y^4}{3^{-2} y^6} \cdot \frac{64 x^{-6} y^9}{27 x^{15}} \cdot \frac{y^6}{\frac{1}{81^2} x^4} = \frac{4 \cdot 64 \cdot y^{4+9+6}}{3^{-2} \cdot 27 \cdot x^{-6-6+4} \cdot \frac{1}{81^2}}

Simplifying the exponents, we get:

4⋅64⋅y193−2⋅27⋅x−8⋅1812=256⋅y1919⋅127⋅x−8⋅16561\frac{4 \cdot 64 \cdot y^{19}}{3^{-2} \cdot 27 \cdot x^{-8} \cdot \frac{1}{81^2}} = \frac{256 \cdot y^{19}}{\frac{1}{9} \cdot \frac{1}{27} \cdot x^{-8} \cdot \frac{1}{6561}}

Simplifying the fractions, we get:

256⋅y1919⋅127⋅x−8⋅16561=256⋅9⋅27⋅x−8⋅6561⋅y19\frac{256 \cdot y^{19}}{\frac{1}{9} \cdot \frac{1}{27} \cdot x^{-8} \cdot \frac{1}{6561}} = 256 \cdot 9 \cdot 27 \cdot x^{-8} \cdot 6561 \cdot y^{19}

Simplifying the constants, we get:

256⋅9⋅27⋅6561=312⋅28⋅318=28⋅330256 \cdot 9 \cdot 27 \cdot 6561 = 3^{12} \cdot 2^{8} \cdot 3^{18} = 2^{8} \cdot 3^{30}

Simplifying the expression, we get:

28⋅330⋅x−8⋅y192^{8} \cdot 3^{30} \cdot x^{-8} \cdot y^{19}

Conclusion

In this article, we have demonstrated how to simplify a complex algebraic expression using various techniques and strategies. We have applied the exponent rules, combined like terms, and simplified fractions to arrive at the final simplified expression. The expression (2x−3y23−1y3)2(4x−2y33x5)3÷(81x−2y−3)−2\left(\frac{2 x^{-3} y^2}{3^{-1} y^3}\right)^2 \left(\frac{4 x^{-2} y^3}{3 x^5}\right)^3 \div \left(\frac{81 x^{-2}}{y^{-3}}\right)^{-2} has been simplified to $2^{8} \cdot 3^{30} \cdot x^{-8} \cdot

Introduction

In our previous article, we demonstrated how to simplify a complex algebraic expression using various techniques and strategies. We applied the exponent rules, combined like terms, and simplified fractions to arrive at the final simplified expression. In this article, we will answer some of the most frequently asked questions related to simplifying algebraic expressions.

Q: What are the basic rules for simplifying algebraic expressions?

A: The basic rules for simplifying algebraic expressions include:

  • Applying the exponent rules, such as (ab)n=anbn(ab)^n = a^n b^n and (am)n=amn(a^m)^n = a^{mn}
  • Combining like terms, such as a+a=2aa + a = 2a and a−a=0a - a = 0
  • Simplifying fractions, such as ab=a⋅1b\frac{a}{b} = a \cdot \frac{1}{b}

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

A: To simplify an expression with multiple variables, you need to apply the exponent rules and combine like terms. For example, if you have the expression 2x2y3+3x2y32x^2y^3 + 3x^2y^3, you can combine the like terms to get 5x2y35x^2y^3.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, such as xx or yy. A constant is a value that does not change, such as 22 or 33.

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

A: To simplify an expression with negative exponents, you need to apply the exponent rule for negative exponents, which states that a−n=1ana^{-n} = \frac{1}{a^n}. For example, if you have the expression x−2x^{-2}, you can rewrite it as 1x2\frac{1}{x^2}.

Q: Can I simplify an expression with a fraction in the denominator?

A: Yes, you can simplify an expression with a fraction in the denominator by multiplying both the numerator and the denominator by the reciprocal of the fraction. For example, if you have the expression 1x+1\frac{1}{x + 1}, you can rewrite it as xx2+x\frac{x}{x^2 + x}.

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

A: To simplify an expression with a radical in the denominator, you need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. For example, if you have the expression 1x+1\frac{1}{\sqrt{x} + 1}, you can rewrite it as x−1x−1\frac{\sqrt{x} - 1}{x - 1}.

Q: Can I simplify an expression with a trigonometric function in the denominator?

A: Yes, you can simplify an expression with a trigonometric function in the denominator by using the trigonometric identities and the exponent rules. For example, if you have the expression 1sin⁡x+1\frac{1}{\sin x + 1}, you can rewrite it as cos⁡x−1sin⁡xcos⁡x\frac{\cos x - 1}{\sin x \cos x}.

Q: How do I simplify an expression with a logarithmic function in the denominator?

A: To simplify an expression with a logarithmic function in the denominator, you need to use the logarithmic identities and the exponent rules. For example, if you have the expression 1log⁡x+1\frac{1}{\log x + 1}, you can rewrite it as elog⁡xelog⁡x+1\frac{e^{\log x}}{e^{\log x + 1}}.

Conclusion

In this article, we have answered some of the most frequently asked questions related to simplifying algebraic expressions. We have demonstrated how to apply the exponent rules, combine like terms, and simplify fractions to arrive at the final simplified expression. By following these techniques and strategies, you can simplify even the most complex algebraic expressions.

Additional Resources

If you need additional help or resources to simplify algebraic expressions, here are some additional resources that you can use:

  • Online algebra calculators and simplifiers
  • Algebra textbooks and workbooks
  • Online algebra courses and tutorials
  • Algebra software and apps

Remember, simplifying algebraic expressions is a skill that takes practice and patience. With consistent practice and review, you can become proficient in simplifying even the most complex expressions.