Simplify The Expression: $\[ \sqrt[3]{x^9 Y^{\frac{1}{3}} Z^{\frac{1}{2}}} \times Y^{\frac{8}{9}} \times \left(2^{-8} X^6 Y^2 Z^{\frac{1}{3}}\right)^{-\frac{1}{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 x9y13z123×y89×(2−8x6y2z13)−12\sqrt[3]{x^9 y^{\frac{1}{3}} z^{\frac{1}{2}}} \times y^{\frac{8}{9}} \times \left(2^{-8} x^6 y^2 z^{\frac{1}{3}}\right)^{-\frac{1}{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. x9y13z123\sqrt[3]{x^9 y^{\frac{1}{3}} z^{\frac{1}{2}}}
  2. y89y^{\frac{8}{9}}
  3. (2−8x6y2z13)−12\left(2^{-8} x^6 y^2 z^{\frac{1}{3}}\right)^{-\frac{1}{2}}

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

Simplifying the First Part

Let's start by simplifying the first part of the expression: x9y13z123\sqrt[3]{x^9 y^{\frac{1}{3}} z^{\frac{1}{2}}}. To simplify this expression, we need to apply the properties of radicals. Specifically, we can use the property that ann=a\sqrt[n]{a^n} = a.

Using this property, we can rewrite the expression as:

x9y13z123=x93×y133×z123\sqrt[3]{x^9 y^{\frac{1}{3}} z^{\frac{1}{2}}} = \sqrt[3]{x^9} \times \sqrt[3]{y^{\frac{1}{3}}} \times \sqrt[3]{z^{\frac{1}{2}}}

Now, we can simplify each of the radicals:

x93=x3\sqrt[3]{x^9} = x^3 y133=y19\sqrt[3]{y^{\frac{1}{3}}} = y^{\frac{1}{9}} z123=z16\sqrt[3]{z^{\frac{1}{2}}} = z^{\frac{1}{6}}

Therefore, the first part of the expression simplifies to:

x3y19z16x^3 y^{\frac{1}{9}} z^{\frac{1}{6}}

Simplifying the Second Part

The second part of the expression is y89y^{\frac{8}{9}}. This expression is already simplified, so we can move on to the next part.

Simplifying the Third Part

The third part of the expression is (2−8x6y2z13)−12\left(2^{-8} x^6 y^2 z^{\frac{1}{3}}\right)^{-\frac{1}{2}}. To simplify this expression, we need to apply the properties of exponents. Specifically, we can use the property that (am)n=amn(a^m)^n = a^{mn}.

Using this property, we can rewrite the expression as:

(2−8x6y2z13)−12=24x−3y−1z−16\left(2^{-8} x^6 y^2 z^{\frac{1}{3}}\right)^{-\frac{1}{2}} = 2^4 x^{-3} y^{-1} z^{-\frac{1}{6}}

Now, we can simplify each of the exponents:

24=162^4 = 16 x−3=1x3x^{-3} = \frac{1}{x^3} y−1=1yy^{-1} = \frac{1}{y} z−16=1z16z^{-\frac{1}{6}} = \frac{1}{z^{\frac{1}{6}}}

Therefore, the third part of the expression simplifies to:

16x3yz16\frac{16}{x^3 y z^{\frac{1}{6}}}

Combining the Simplified Parts

Now that we have simplified each of the parts of the expression, we can combine them to get the final simplified expression.

The first part of the expression is x3y19z16x^3 y^{\frac{1}{9}} z^{\frac{1}{6}}.

The second part of the expression is y89y^{\frac{8}{9}}.

The third part of the expression is 16x3yz16\frac{16}{x^3 y z^{\frac{1}{6}}}.

To combine these parts, we can multiply them together:

x3y19z16×y89×16x3yz16x^3 y^{\frac{1}{9}} z^{\frac{1}{6}} \times y^{\frac{8}{9}} \times \frac{16}{x^3 y z^{\frac{1}{6}}}

Now, we can simplify the expression by combining like terms:

x3y19z16×y89×16x3yz16=16y89+19=16y1=16yx^3 y^{\frac{1}{9}} z^{\frac{1}{6}} \times y^{\frac{8}{9}} \times \frac{16}{x^3 y z^{\frac{1}{6}}} = 16 y^{\frac{8}{9} + \frac{1}{9}} = 16 y^1 = 16y

Therefore, the final simplified expression is:

16y16y

Conclusion

In this article, we have demonstrated how to simplify a complex algebraic expression using various techniques and strategies. We have broken down the expression into its individual parts and simplified each of them using properties of radicals, exponents, and variables. By combining the simplified parts, we have arrived at the final simplified expression. This example illustrates the importance of careful algebraic manipulation in simplifying complex expressions and reveals the underlying structure of the expression.

Frequently Asked Questions

  • Q: What is the main goal of simplifying an algebraic expression? A: The main goal of simplifying an algebraic expression is to reveal its underlying structure and make it easier to work with.
  • Q: What are some common techniques used to simplify algebraic expressions? A: Some common techniques used to simplify algebraic expressions include applying properties of radicals, exponents, and variables, as well as combining like terms.
  • Q: How do I know when an algebraic expression is simplified? A: An algebraic expression is considered simplified when it cannot be further reduced using the properties of radicals, exponents, and variables.

Further Reading

References

Introduction

In our previous article, we demonstrated how to simplify a complex algebraic expression using various techniques and strategies. We broke down the expression into its individual parts and simplified each of them using properties of radicals, exponents, and variables. By combining the simplified parts, we arrived at the final simplified expression. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the main goal of simplifying an algebraic expression?

A: The main goal of simplifying an algebraic expression is to reveal its underlying structure and make it easier to work with.

Q: What are some common techniques used to simplify algebraic expressions?

A: Some common techniques used to simplify algebraic expressions include applying properties of radicals, exponents, and variables, as well as combining like terms.

Q: How do I know when an algebraic expression is simplified?

A: An algebraic expression is considered simplified when it cannot be further reduced using the properties of radicals, exponents, and variables.

Q: What is the difference between simplifying an algebraic expression and solving an equation?

A: Simplifying an algebraic expression involves reducing the expression to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.

Q: Can I simplify an algebraic expression that contains a variable with a negative exponent?

A: Yes, you can simplify an algebraic expression that contains a variable with a negative exponent by applying the property that a−m=1ama^{-m} = \frac{1}{a^m}.

Q: How do I simplify an algebraic expression that contains a radical with a variable in the radicand?

A: To simplify an algebraic expression that contains a radical with a variable in the radicand, you can apply the property that ann=a\sqrt[n]{a^n} = a.

Q: Can I simplify an algebraic expression that contains a fraction with a variable in the numerator or denominator?

A: Yes, you can simplify an algebraic expression that contains a fraction with a variable in the numerator or denominator by applying the properties of fractions, such as multiplying the numerator and denominator by the same value to eliminate the variable.

Q: How do I simplify an algebraic expression that contains multiple variables?

A: To simplify an algebraic expression that contains multiple variables, you can apply the properties of variables, such as combining like terms and applying the distributive property.

Q: Can I simplify an algebraic expression that contains a trigonometric function?

A: Yes, you can simplify an algebraic expression that contains a trigonometric function by applying the properties of trigonometric functions, such as the Pythagorean identity.

Q: How do I simplify an algebraic expression that contains a logarithmic function?

A: To simplify an algebraic expression that contains a logarithmic function, you can apply the properties of logarithmic functions, such as the power rule and the product rule.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying algebraic expressions. We have covered topics such as the main goal of simplifying an algebraic expression, common techniques used to simplify algebraic expressions, and how to simplify expressions that contain variables with negative exponents, radicals with variables in the radicand, fractions with variables in the numerator or denominator, multiple variables, trigonometric functions, and logarithmic functions. By understanding these concepts and techniques, you can simplify complex algebraic expressions and reveal their underlying structure.

Frequently Asked Questions

  • Q: What is the main goal of simplifying an algebraic expression? A: The main goal of simplifying an algebraic expression is to reveal its underlying structure and make it easier to work with.
  • Q: What are some common techniques used to simplify algebraic expressions? A: Some common techniques used to simplify algebraic expressions include applying properties of radicals, exponents, and variables, as well as combining like terms.
  • Q: How do I know when an algebraic expression is simplified? A: An algebraic expression is considered simplified when it cannot be further reduced using the properties of radicals, exponents, and variables.

Further Reading

References