Simplify: $\sqrt{\frac{378 X^7 Y^5}{363 X^4 Y^7}}$Provide Your Answer Below:

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Understanding the Problem

The given problem involves simplifying a radical expression, which is a mathematical expression that contains a square root. The expression is $\sqrt{\frac{378 x^7 y^5}{363 x^4 y^7}}$. To simplify this expression, we need to use the properties of radicals and exponents.

Breaking Down the Expression

The given expression can be broken down into two parts: the numerator and the denominator. The numerator is $378 x^7 y^5$ and the denominator is $363 x^4 y^7$. We can simplify each part separately before combining them.

Simplifying the Numerator

The numerator is $378 x^7 y^5$. We can start by factoring out the greatest common factor (GCF) of the coefficients and the variables. The GCF of the coefficients is 18, and the GCF of the variables is $x^4 y^3$. Therefore, we can rewrite the numerator as:

$$378 x^7 y^5 = 18 \cdot 21 \cdot x^4 \cdot x^3 \cdot y^3 \cdot y^2$$

Simplifying the Denominator

The denominator is $363 x^4 y^7$. We can start by factoring out the greatest common factor (GCF) of the coefficients and the variables. The GCF of the coefficients is 3, and the GCF of the variables is $x^4 y^3$. Therefore, we can rewrite the denominator as:

$$363 x^4 y^7 = 3 \cdot 121 \cdot x^4 \cdot y^3 \cdot y^4$$

Combining the Numerator and Denominator

Now that we have simplified the numerator and the denominator, we can combine them to get the simplified expression:

$$\sqrt{\frac{378 x^7 y^5}{363 x^4 y^7}} = \sqrt{\frac{18 \cdot 21 \cdot x^4 \cdot x^3 \cdot y^3 \cdot y^2}{3 \cdot 121 \cdot x^4 \cdot y^3 \cdot y^4}}$$

Canceling Out Common Factors

We can cancel out the common factors in the numerator and the denominator. The common factors are $x^4$, $y^3$, and 3. Therefore, we can cancel them out to get:

$$\sqrt{\frac{18 \cdot 21 \cdot x^3 \cdot y^2}{121 \cdot y^4}}$$

Simplifying the Expression

Now that we have canceled out the common factors, we can simplify the expression further. We can start by simplifying the coefficients. The coefficient 18 can be rewritten as $2 \cdot 3^2$, and the coefficient 21 can be rewritten as $3 \cdot 7$. Therefore, we can rewrite the expression as:

$$\sqrt{\frac{2 \cdot 3^2 \cdot 3 \cdot 7 \cdot x^3 \cdot y^2}{121 \cdot y^4}}$$

Canceling Out Common Factors Again

We can cancel out the common factors in the numerator and the denominator. The common factors are $3^2$, $3$, and $y^2$. Therefore, we can cancel them out to get:

$$\sqrt{\frac{2 \cdot 7 \cdot x^3}{121 \cdot y^2}}$$

Simplifying the Expression Further

Now that we have canceled out the common factors, we can simplify the expression further. We can start by simplifying the coefficients. The coefficient 121 can be rewritten as $11^2$. Therefore, we can rewrite the expression as:

$$\sqrt{\frac{2 \cdot 7 \cdot x^3}{11^2 \cdot y^2}}$$

Final Simplification

We can simplify the expression further by canceling out the common factors in the numerator and the denominator. The common factors are $2$ and $y^2$. Therefore, we can cancel them out to get:

$$\sqrt{\frac{7 \cdot x^3}{11^2 \cdot y^2}}$$

Conclusion

The simplified expression is $\sqrt{\frac{7 \cdot x^3}{11^2 \cdot y^2}}$. This expression cannot be simplified further.

Final Answer

The final answer is $\boxed{\sqrt{\frac{7x^3}{121y^2}}}$

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Understanding the Problem

The given problem involves simplifying a radical expression, which is a mathematical expression that contains a square root. The expression is $\sqrt{\frac{378 x^7 y^5}{363 x^4 y^7}}$. To simplify this expression, we need to use the properties of radicals and exponents.

Q&A

Q: What is the first step in simplifying the given expression?

A: The first step in simplifying the given expression is to break it down into two parts: the numerator and the denominator. The numerator is $378 x^7 y^5$ and the denominator is $363 x^4 y^7$.

Q: How do we simplify the numerator?

A: We can simplify the numerator by factoring out the greatest common factor (GCF) of the coefficients and the variables. The GCF of the coefficients is 18, and the GCF of the variables is $x^4 y^3$. Therefore, we can rewrite the numerator as:

$$378 x^7 y^5 = 18 \cdot 21 \cdot x^4 \cdot x^3 \cdot y^3 \cdot y^2$$

Q: How do we simplify the denominator?

A: We can simplify the denominator by factoring out the greatest common factor (GCF) of the coefficients and the variables. The GCF of the coefficients is 3, and the GCF of the variables is $x^4 y^3$. Therefore, we can rewrite the denominator as:

$$363 x^4 y^7 = 3 \cdot 121 \cdot x^4 \cdot y^3 \cdot y^4$$

Q: How do we combine the numerator and denominator?

A: We can combine the numerator and denominator by dividing the numerator by the denominator. This gives us:

$$\sqrt{\frac{378 x^7 y^5}{363 x^4 y^7}} = \sqrt{\frac{18 \cdot 21 \cdot x^4 \cdot x^3 \cdot y^3 \cdot y^2}{3 \cdot 121 \cdot x^4 \cdot y^3 \cdot y^4}}$$

Q: How do we cancel out common factors?

A: We can cancel out common factors in the numerator and denominator. The common factors are $x^4$, $y^3$, and 3. Therefore, we can cancel them out to get:

$$\sqrt{\frac{18 \cdot 21 \cdot x^3 \cdot y^2}{121 \cdot y^4}}$$

Q: How do we simplify the expression further?

A: We can simplify the expression further by canceling out common factors in the numerator and denominator. The common factors are $3^2$, $3$, and $y^2$. Therefore, we can cancel them out to get:

$$\sqrt{\frac{2 \cdot 7 \cdot x^3}{121 \cdot y^2}}$$

Q: What is the final simplified expression?

A: The final simplified expression is $\sqrt{\frac{7 \cdot x^3}{11^2 \cdot y^2}}$.

Conclusion

Simplifying a radical expression involves breaking it down into two parts, simplifying each part, and then combining them. We can use the properties of radicals and exponents to simplify the expression. In this case, we simplified the expression by canceling out common factors and simplifying the coefficients.

Final Answer

The final answer is $\boxed{\sqrt{\frac{7x^3}{121y^2}}}$