Simplify The Expression:$\[ \sqrt[4]{\frac{405 X^3 Y^3}{5 X^{-1} Y}} \\]

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Introduction

Simplifying expressions involving exponents and roots is a crucial skill in mathematics, particularly in algebra and calculus. In this article, we will focus on simplifying the given expression, which involves a fourth root and fractions with exponents. We will break down the expression step by step, using the properties of exponents and roots to simplify it.

Understanding the Expression

The given expression is $\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}}$. To simplify this expression, we need to understand the properties of exponents and roots. The fourth root of a number is a number that, when raised to the fourth power, gives the original number. In mathematical notation, this is represented as $\sqrt[4]{x} = y \iff x = y^4$.

Simplifying the Fraction

The first step in simplifying the expression is to simplify the fraction inside the fourth root. We can do this by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 405 and 5 is 5. So, we can simplify the fraction as follows:

$\frac{405 x^3 y^3}{5 x^{-1} y} = \frac{405}{5} \cdot \frac{x^3}{x^{-1}} \cdot \frac{y^3}{y}$

Simplifying the Exponents

Now, we can simplify the exponents in the expression. We know that $x^3 \cdot x^{-1} = x^{3-1} = x^2$. Similarly, $y^3 \cdot y^{-1} = y^{3-1} = y^2$. So, the expression becomes:

$\frac{405}{5} \cdot x^2 \cdot y^2$

Simplifying the Coefficients

The next step is to simplify the coefficients in the expression. We can do this by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 405 and 5 is 5. So, we can simplify the coefficients as follows:

$\frac{405}{5} = \frac{81}{1}$

Simplifying the Expression

Now, we can simplify the expression by combining the simplified coefficients and exponents:

$\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}} = \sqrt[4]{81 x^2 y^2}$

Using the Properties of Roots

To simplify the expression further, we can use the properties of roots. We know that $\sqrt[n]{x^n} = x$. So, we can simplify the expression as follows:

$\sqrt[4]{81 x^2 y^2} = \sqrt[4]{(3^4) x^2 y^2}$

Simplifying the Expression

Now, we can simplify the expression by using the properties of exponents. We know that $x^2 \cdot y^2 = (xy)^2$. So, the expression becomes:

$\sqrt[4]{(3^4) (xy)^2}$

Simplifying the Expression

Finally, we can simplify the expression by using the properties of roots. We know that $\sqrt[n]{x^n} = x$. So, the expression becomes:

$\sqrt[4]{(3^4) (xy)^2} = 3xy$

The final answer is $\boxed{3xy}$.

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Introduction

In our previous article, we simplified the expression $\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}}$ step by step, using the properties of exponents and roots. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of this expression.

Q: What is the fourth root of a number?

A: The fourth root of a number is a number that, when raised to the fourth power, gives the original number. In mathematical notation, this is represented as $\sqrt[4]{x} = y \iff x = y^4$.

Q: How do I simplify a fraction with exponents?

A: To simplify a fraction with exponents, you can divide the numerator and denominator by their greatest common divisor (GCD). You can also use the properties of exponents to simplify the expression.

Q: What is the property of exponents that states $x^m \cdot x^n = x^{m+n}$?

A: This property is known as the product of powers property. It states that when you multiply two numbers with the same base, you can add their exponents.

Q: How do I simplify an expression with a fourth root?

A: To simplify an expression with a fourth root, you can use the properties of roots. You can also use the properties of exponents to simplify the expression.

Q: What is the final answer to the expression $\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}}$?

A: The final answer to the expression $\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}}$ is $3xy$.

Q: Can I use a calculator to simplify the expression $\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}}$?

A: Yes, you can use a calculator to simplify the expression $\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}}$. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions?

A: To apply the properties of exponents and roots to simplify expressions, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with negative exponents?

A: Yes, you can use the properties of exponents and roots to simplify expressions with negative exponents. You can also use the properties of exponents to simplify expressions with fractional exponents.

Q: How do I simplify expressions with multiple roots?

A: To simplify expressions with multiple roots, you can use the properties of roots. You can also use the properties of exponents to simplify the expression.

Q: Can I use the properties of exponents and roots to simplify expressions with variables?

A: Yes, you can use the properties of exponents and roots to simplify expressions with variables. You can also use the properties of exponents to simplify expressions with fractional exponents.

Q: How do I apply the properties of exponents and roots to simplify expressions with multiple variables?

A: To apply the properties of exponents and roots to simplify expressions with multiple variables, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with complex numbers?

A: Yes, you can use the properties of exponents and roots to simplify expressions with complex numbers. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with trigonometric functions?

A: To apply the properties of exponents and roots to simplify expressions with trigonometric functions, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with logarithmic functions?

A: Yes, you can use the properties of exponents and roots to simplify expressions with logarithmic functions. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with exponential functions?

A: To apply the properties of exponents and roots to simplify expressions with exponential functions, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with polynomial functions?

A: Yes, you can use the properties of exponents and roots to simplify expressions with polynomial functions. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with rational functions?

A: To apply the properties of exponents and roots to simplify expressions with rational functions, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with trigonometric identities?

A: Yes, you can use the properties of exponents and roots to simplify expressions with trigonometric identities. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with logarithmic identities?

A: To apply the properties of exponents and roots to simplify expressions with logarithmic identities, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with exponential identities?

A: Yes, you can use the properties of exponents and roots to simplify expressions with exponential identities. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with polynomial identities?

A: To apply the properties of exponents and roots to simplify expressions with polynomial identities, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with rational identities?

A: Yes, you can use the properties of exponents and roots to simplify expressions with rational identities. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with trigonometric identities?

A: To apply the properties of exponents and roots to simplify expressions with trigonometric identities, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with logarithmic identities?

A: Yes, you can use the properties of exponents and roots to simplify expressions with logarithmic identities. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with exponential identities?

A: To apply the properties of exponents and roots to simplify expressions with exponential identities, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Q: Can I use the properties of exponents and roots to simplify expressions with polynomial identities?

A: Yes, you can use the properties of exponents and roots to simplify expressions with polynomial identities. However, it's always a good idea to understand the underlying math and simplify the expression step by step.

Q: How do I apply the properties of exponents and roots to simplify expressions with rational identities?

A: To apply the properties of exponents and roots to simplify expressions with rational identities, you need to understand the properties of exponents and roots. You can then use these properties to simplify the expression step by step.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to the simplification of the expression $\sqrt[4]{\frac{405 x^3 y^3}{5 x^{-1} y}}$. We have also provided some additional information on how to apply the properties of exponents and roots to simplify expressions with various types of functions.