Order The Simplification Steps Of The Expression Below Using The Properties Of Rational Exponents.Expression: $\sqrt[4]{567 X^9 Y^{11}}$1. $(567 X^9 Y^{11})^{\frac{1}{4}}$2. $(81 \cdot 7)^{\frac{1}{4}} \cdot X^{\frac{9}{4}} \cdot

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Introduction

Rational exponents are a powerful tool in algebra, allowing us to simplify complex expressions and solve equations. In this article, we will explore the properties of rational exponents and use them to simplify the expression 567x9y114\sqrt[4]{567 x^9 y^{11}}. We will examine two different approaches to simplifying this expression and determine the correct order of steps.

Understanding Rational Exponents

Rational exponents are exponents that can be expressed as a fraction, where the numerator is a positive integer and the denominator is a positive integer. For example, 12\frac{1}{2} and 34\frac{3}{4} are rational exponents. Rational exponents can be used to simplify expressions and solve equations.

Properties of Rational Exponents

There are several properties of rational exponents that we will use to simplify the expression 567x9y114\sqrt[4]{567 x^9 y^{11}}. These properties include:

  • Product of Powers: When we multiply two powers with the same base, we add their exponents. For example, xaâ‹…xb=xa+bx^a \cdot x^b = x^{a+b}.
  • Power of a Power: When we raise a power to a power, we multiply their exponents. For example, (xa)b=xab(x^a)^b = x^{ab}.
  • Root of a Power: When we take the root of a power, we divide the exponent by the index of the root. For example, xan=xan\sqrt[n]{x^a} = x^{\frac{a}{n}}.

Approach 1: Simplifying the Expression Using the Product of Powers Property

Our first approach to simplifying the expression 567x9y114\sqrt[4]{567 x^9 y^{11}} is to use the product of powers property. We can rewrite the expression as:

567x9y114=(567x9y11)14\sqrt[4]{567 x^9 y^{11}} = (567 x^9 y^{11})^{\frac{1}{4}}

Using the product of powers property, we can rewrite this expression as:

(567x9y11)14=56714â‹…x94â‹…y114(567 x^9 y^{11})^{\frac{1}{4}} = 567^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}

Approach 2: Simplifying the Expression Using the Root of a Power Property

Our second approach to simplifying the expression 567x9y114\sqrt[4]{567 x^9 y^{11}} is to use the root of a power property. We can rewrite the expression as:

567x9y114=5674â‹…x94â‹…y114\sqrt[4]{567 x^9 y^{11}} = \sqrt[4]{567} \cdot \sqrt[4]{x^9} \cdot \sqrt[4]{y^{11}}

Using the root of a power property, we can rewrite this expression as:

5674â‹…x94â‹…y114=8114â‹…714â‹…x94â‹…y114\sqrt[4]{567} \cdot \sqrt[4]{x^9} \cdot \sqrt[4]{y^{11}} = 81^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}

Ordering the Simplification Steps

Now that we have simplified the expression using both approaches, we can determine the correct order of steps. Our first approach used the product of powers property, while our second approach used the root of a power property.

In general, when simplifying an expression using rational exponents, we should follow these steps:

  1. Rewrite the expression as a power: Rewrite the expression as a power using the product of powers property.
  2. Simplify the power: Simplify the power using the properties of exponents.
  3. Rewrite the expression as a root: Rewrite the expression as a root using the root of a power property.
  4. Simplify the root: Simplify the root using the properties of exponents.

Conclusion

In this article, we have explored the properties of rational exponents and used them to simplify the expression 567x9y114\sqrt[4]{567 x^9 y^{11}}. We have examined two different approaches to simplifying this expression and determined the correct order of steps. By following these steps, we can simplify complex expressions and solve equations using rational exponents.

Final Answer

Introduction

In our previous article, we explored the properties of rational exponents and used them to simplify the expression 567x9y114\sqrt[4]{567 x^9 y^{11}}. We also determined the correct order of steps for simplifying expressions using rational exponents. In this article, we will answer some common questions about simplifying expressions with rational exponents.

Q: What is the difference between a rational exponent and a fractional exponent?

A: A rational exponent is an exponent that can be expressed as a fraction, where the numerator is a positive integer and the denominator is a positive integer. For example, 12\frac{1}{2} and 34\frac{3}{4} are rational exponents. A fractional exponent, on the other hand, is an exponent that is a fraction, but the numerator and denominator may be negative or zero. For example, −12-\frac{1}{2} and 04\frac{0}{4} are fractional exponents.

Q: How do I simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, you can use the following steps:

  1. Rewrite the expression as a power: Rewrite the expression as a power using the product of powers property.
  2. Simplify the power: Simplify the power using the properties of exponents.
  3. Rewrite the expression as a root: Rewrite the expression as a root using the root of a power property.
  4. Simplify the root: Simplify the root using the properties of exponents.

Q: What is the product of powers property?

A: The product of powers property states that when we multiply two powers with the same base, we add their exponents. For example, xaâ‹…xb=xa+bx^a \cdot x^b = x^{a+b}.

Q: What is the root of a power property?

A: The root of a power property states that when we take the root of a power, we divide the exponent by the index of the root. For example, xan=xan\sqrt[n]{x^a} = x^{\frac{a}{n}}.

Q: How do I simplify an expression with a negative rational exponent?

A: To simplify an expression with a negative rational exponent, you can use the following steps:

  1. Rewrite the expression as a power: Rewrite the expression as a power using the product of powers property.
  2. Simplify the power: Simplify the power using the properties of exponents.
  3. Rewrite the expression as a root: Rewrite the expression as a root using the root of a power property.
  4. Simplify the root: Simplify the root using the properties of exponents.

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

A: A rational exponent is an exponent that can be expressed as a fraction, where the numerator is a positive integer and the denominator is a positive integer. A radical, on the other hand, is a symbol that represents a root. For example, x\sqrt{x} represents the square root of xx.

Q: How do I simplify an expression with a rational exponent and a radical?

A: To simplify an expression with a rational exponent and a radical, you can use the following steps:

  1. Rewrite the expression as a power: Rewrite the expression as a power using the product of powers property.
  2. Simplify the power: Simplify the power using the properties of exponents.
  3. Rewrite the expression as a root: Rewrite the expression as a root using the root of a power property.
  4. Simplify the root: Simplify the root using the properties of exponents.

Conclusion

In this article, we have answered some common questions about simplifying expressions with rational exponents. We have also provided some tips and tricks for simplifying expressions with rational exponents. By following these steps and using the properties of exponents, you can simplify complex expressions and solve equations using rational exponents.

Final Answer

The final answer is: 1\boxed{1}