Order The Simplification Steps Of The Expression Below Using The Properties Of Rational Exponents.$\sqrt[4]{567 X^9 Y^{11}}$\[ \begin{array}{|c|} \hline (81 \cdot 7)^{\frac{1}{2}} \cdot X^{\frac{2}{2}} \cdot Y^{\frac{4}{4}} \\ \hline 3^1

Understanding Rational Exponents

Rational exponents are a way to express roots and powers of numbers using fractions. They are a powerful tool in algebra and are used to simplify expressions and solve equations. In this article, we will use the properties of rational exponents to simplify the expression $\sqrt[4]{567 x^9 y^{11}}$.

Breaking Down the Expression

The given expression is $\sqrt[4]{567 x^9 y^{11}}$. To simplify this expression, we need to break it down into its prime factors. We can start by factoring the number 567.

Factoring 567

567 can be factored as $3^3 \cdot 7 \cdot 3$. Therefore, we can rewrite the expression as $\sqrt[4]{3^3 \cdot 7 \cdot 3 x^9 y^{11}}$.

Applying the Properties of Rational Exponents

Now that we have factored the number 567, we can apply the properties of rational exponents to simplify the expression. The properties of rational exponents state that:

• $\sqrt[n]{a} = a^{\frac{1}{n}}$
• $(a^m)^n = a^{m \cdot n}$
• $a^m \cdot a^n = a^{m + n}$

Using these properties, we can simplify the expression as follows:

$\sqrt[4]{3^3 \cdot 7 \cdot 3 x^9 y^{11}} = (3^3 \cdot 7 \cdot 3)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}$

Simplifying the Expression

Now that we have applied the properties of rational exponents, we can simplify the expression further. We can start by simplifying the term $(3^3 \cdot 7 \cdot 3)^{\frac{1}{4}}$.

Simplifying the Term $(3^3 \cdot 7 \cdot 3)^{\frac{1}{4}}$

Using the property $(a^m)^n = a^{m \cdot n}$, we can rewrite the term as $(3^3 \cdot 7 \cdot 3)^{\frac{1}{4}} = (3^3)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 3^{\frac{1}{4}}$.

Simplifying the Terms $(3^3)^{\frac{1}{4}}$, $7^{\frac{1}{4}}$, and $3^{\frac{1}{4}}$

Using the property $\sqrt[n]{a} = a^{\frac{1}{n}}$, we can simplify the terms as follows:

• $(3^3)^{\frac{1}{4}} = 3^{\frac{3}{4}}$
• $7^{\frac{1}{4}} = \sqrt[4]{7}$
• $3^{\frac{1}{4}} = \sqrt[4]{3}$

Simplifying the Expression Further

Now that we have simplified the terms, we can simplify the expression further. We can start by combining the terms $3^{\frac{3}{4}}$, $\sqrt[4]{7}$, and $\sqrt[4]{3}$.

Combining the Terms $3^{\frac{3}{4}}$, $\sqrt[4]{7}$, and $\sqrt[4]{3}$

Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$3^{\frac{3}{4}} \cdot \sqrt[4]{7} \cdot \sqrt[4]{3} = 3^{\frac{3}{4} + \frac{1}{4} + \frac{1}{4}} \cdot \sqrt[4]{7 \cdot 3}$

Simplifying the Term $3^{\frac{3}{4} + \frac{1}{4} + \frac{1}{4}}$

Using the property $a^m \cdot a^n = a^{m + n}$, we can simplify the term as follows:

$3^{\frac{3}{4} + \frac{1}{4} + \frac{1}{4}} = 3^{\frac{3}{4} + \frac{2}{4}} = 3^{\frac{5}{4}}$

Simplifying the Term $\sqrt[4]{7 \cdot 3}$

Using the property $\sqrt[n]{a} = a^{\frac{1}{n}}$, we can simplify the term as follows:

$\sqrt[4]{7 \cdot 3} = (7 \cdot 3)^{\frac{1}{4}} = 3^{\frac{1}{4}} \cdot 7^{\frac{1}{4}}$

Simplifying the Expression Further

Now that we have simplified the terms, we can simplify the expression further. We can start by combining the terms $3^{\frac{5}{4}}$ and $3^{\frac{1}{4}} \cdot 7^{\frac{1}{4}}$.

Combining the Terms $3^{\frac{5}{4}}$ and $3^{\frac{1}{4}} \cdot 7^{\frac{1}{4}}$

Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$3^{\frac{5}{4}} \cdot 3^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} = 3^{\frac{5}{4} + \frac{1}{4}} \cdot 7^{\frac{1}{4}}$

Simplifying the Term $3^{\frac{5}{4} + \frac{1}{4}}$

Using the property $a^m \cdot a^n = a^{m + n}$, we can simplify the term as follows:

$3^{\frac{5}{4} + \frac{1}{4}} = 3^{\frac{6}{4}} = 3^{\frac{3}{2}}$

Simplifying the Expression Further

Now that we have simplified the terms, we can simplify the expression further. We can start by combining the terms $3^{\frac{3}{2}}$ and $7^{\frac{1}{4}}$.

Combining the Terms $3^{\frac{3}{2}}$ and $7^{\frac{1}{4}}$

Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}} = 3^{\frac{3}{2}} \cdot (7^{\frac{1}{2}})^{\frac{1}{2}}$

Simplifying the Term $3^{\frac{3}{2}} \cdot (7^{\frac{1}{2}})^{\frac{1}{2}}$

Using the property $(a^m)^n = a^{m \cdot n}$, we can simplify the term as follows:

$3^{\frac{3}{2}} \cdot (7^{\frac{1}{2}})^{\frac{1}{2}} = 3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}}$

Simplifying the Expression Further

Now that we have simplified the terms, we can simplify the expression further. We can start by combining the terms $x^{\frac{9}{4}}$ and $y^{\frac{11}{4}}$.

Combining the Terms $x^{\frac{9}{4}}$ and $y^{\frac{11}{4}}$

Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} = x^{\frac{9}{4} + \frac{11}{4}} \cdot y^{\frac{11}{4}}$

Simplifying the Term $x^{\frac{9}{4} + \frac{11}{4}}$

Using the property $a^m \cdot a^n = a^{m + n}$, we can simplify the term as follows:

$x^{\frac{9}{4} + \frac{11}{4}} = x^{\frac{20}{4}} = x^{\frac{5}{1}}$

Simplifying the Expression Further

Now that we have simplified the terms, we can simplify the expression further. We can start by combining the terms $3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}}$ and $x^{\frac{5}{1}}$.

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Q: What are rational exponents?

A: Rational exponents are a way to express roots and powers of numbers using fractions. They are a powerful tool in algebra and are used to simplify expressions and solve equations.

Q: How do I simplify the expression $\sqrt[4]{567 x^9 y^{11}}$ using rational exponents?

A: To simplify the expression, we need to break it down into its prime factors. We can start by factoring the number 567. Then, we can apply the properties of rational exponents to simplify the expression.

Q: What are the properties of rational exponents?

A: The properties of rational exponents state that:

• $\sqrt[n]{a} = a^{\frac{1}{n}}$
• $(a^m)^n = a^{m \cdot n}$
• $a^m \cdot a^n = a^{m + n}$

Q: How do I apply the properties of rational exponents to simplify the expression?

A: We can start by simplifying the term $(3^3 \cdot 7 \cdot 3)^{\frac{1}{4}}$. Using the property $(a^m)^n = a^{m \cdot n}$, we can rewrite the term as $(3^3)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot 3^{\frac{1}{4}}$.

Q: How do I simplify the terms $(3^3)^{\frac{1}{4}}$, $7^{\frac{1}{4}}$, and $3^{\frac{1}{4}}$?

A: Using the property $\sqrt[n]{a} = a^{\frac{1}{n}}$, we can simplify the terms as follows:

• $(3^3)^{\frac{1}{4}} = 3^{\frac{3}{4}}$
• $7^{\frac{1}{4}} = \sqrt[4]{7}$
• $3^{\frac{1}{4}} = \sqrt[4]{3}$

Q: How do I combine the terms $3^{\frac{3}{4}}$, $\sqrt[4]{7}$, and $\sqrt[4]{3}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$3^{\frac{3}{4}} \cdot \sqrt[4]{7} \cdot \sqrt[4]{3} = 3^{\frac{3}{4} + \frac{1}{4} + \frac{1}{4}} \cdot \sqrt[4]{7 \cdot 3}$

Q: How do I simplify the term $3^{\frac{3}{4} + \frac{1}{4} + \frac{1}{4}}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can simplify the term as follows:

$3^{\frac{3}{4} + \frac{1}{4} + \frac{1}{4}} = 3^{\frac{5}{4}}$

Q: How do I simplify the term $\sqrt[4]{7 \cdot 3}$?

A: Using the property $\sqrt[n]{a} = a^{\frac{1}{n}}$, we can simplify the term as follows:

$\sqrt[4]{7 \cdot 3} = (7 \cdot 3)^{\frac{1}{4}} = 3^{\frac{1}{4}} \cdot 7^{\frac{1}{4}}$

Q: How do I combine the terms $3^{\frac{5}{4}}$ and $3^{\frac{1}{4}} \cdot 7^{\frac{1}{4}}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$3^{\frac{5}{4}} \cdot 3^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} = 3^{\frac{5}{4} + \frac{1}{4}} \cdot 7^{\frac{1}{4}}$

Q: How do I simplify the term $3^{\frac{5}{4} + \frac{1}{4}}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can simplify the term as follows:

$3^{\frac{5}{4} + \frac{1}{4}} = 3^{\frac{6}{4}} = 3^{\frac{3}{2}}$

Q: How do I combine the terms $3^{\frac{3}{2}}$ and $7^{\frac{1}{4}}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}} = 3^{\frac{3}{2}} \cdot (7^{\frac{1}{2}})^{\frac{1}{2}}$

Q: How do I simplify the term $3^{\frac{3}{2}} \cdot (7^{\frac{1}{2}})^{\frac{1}{2}}$?

A: Using the property $(a^m)^n = a^{m \cdot n}$, we can simplify the term as follows:

$3^{\frac{3}{2}} \cdot (7^{\frac{1}{2}})^{\frac{1}{2}} = 3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}}$

Q: How do I combine the terms $x^{\frac{9}{4}}$ and $y^{\frac{11}{4}}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} = x^{\frac{9}{4} + \frac{11}{4}} \cdot y^{\frac{11}{4}}$

Q: How do I simplify the term $x^{\frac{9}{4} + \frac{11}{4}}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can simplify the term as follows:

$x^{\frac{9}{4} + \frac{11}{4}} = x^{\frac{20}{4}} = x^{\frac{5}{1}}$

Q: How do I combine the terms $3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}}$ and $x^{\frac{5}{1}}$?

A: Using the property $a^m \cdot a^n = a^{m + n}$, we can combine the terms as follows:

$3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{5}{1}} = 3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{5}{1}}$

The final answer is $\boxed{3^{\frac{3}{2}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{5}{1}} \cdot y^{\frac{11}{4}}}$