Simplify The Expression:${ -7 \sqrt[3]{250 X^{12} Y^7 Z^{20}} }$

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Understanding the Problem

When dealing with expressions involving radicals, it's essential to simplify them by factoring out perfect cubes and other factors that can be extracted from the radicand. In this case, we're given the expression $-7 \sqrt[3]{250 x^{12} y^7 z^{20}}$, and our goal is to simplify it.

Breaking Down the Radicand

To simplify the expression, we need to break down the radicand into its prime factors. The radicand is $250 x^{12} y^7 z^{20}$. We can start by factoring out the greatest perfect cube that divides each of the factors.

Factoring Out Perfect Cubes

We can factor out perfect cubes from the radicand as follows:

• $250 = 2 \cdot 5^3 \cdot 2$
• $x^{12} = (x^3)^4$
• $y^7 = (y^3)^2 \cdot y$
• $z^{20} = (z^3)^6 \cdot z^2$

Simplifying the Expression

Now that we've factored out perfect cubes, we can rewrite the radicand as:

$250 x^{12} y^7 z^{20} = 2 \cdot 5^3 \cdot 2 \cdot (x^3)^4 \cdot (y^3)^2 \cdot y \cdot (z^3)^6 \cdot z^2$

We can simplify this expression by combining like terms:

$250 x^{12} y^7 z^{20} = 2^2 \cdot 5^3 \cdot (x^3)^4 \cdot (y^3)^2 \cdot (z^3)^6 \cdot y \cdot z^2$

Applying the Properties of Radicals

Now that we've simplified the radicand, we can apply the properties of radicals to simplify the expression. Specifically, we can use the property that $\sqrt[3]{a^3} = a$ to simplify the expression.

Simplifying the Expression Further

Using the property of radicals, we can simplify the expression as follows:

$-7 \sqrt[3]{250 x^{12} y^7 z^{20}} = -7 \sqrt[3]{2^2 \cdot 5^3 \cdot (x^3)^4 \cdot (y^3)^2 \cdot (z^3)^6 \cdot y \cdot z^2}$

We can simplify this expression further by applying the property of radicals:

$-7 \sqrt[3]{2^2 \cdot 5^3 \cdot (x^3)^4 \cdot (y^3)^2 \cdot (z^3)^6 \cdot y \cdot z^2} = -7 \cdot 2 \cdot 5 \cdot x^4 \cdot y^2 \cdot z^6 \cdot \sqrt[3]{y \cdot z^2}$

Final Simplification

The final simplified expression is:

$-7 \cdot 2 \cdot 5 \cdot x^4 \cdot y^2 \cdot z^6 \cdot \sqrt[3]{y \cdot z^2}$

Conclusion

In this article, we simplified the expression $-7 \sqrt[3]{250 x^{12} y^7 z^{20}}$ by factoring out perfect cubes and applying the properties of radicals. We broke down the radicand into its prime factors, factored out perfect cubes, and simplified the expression further by applying the properties of radicals. The final simplified expression is $-7 \cdot 2 \cdot 5 \cdot x^4 \cdot y^2 \cdot z^6 \cdot \sqrt[3]{y \cdot z^2}$.

Frequently Asked Questions

• What is the greatest perfect cube that divides each of the factors in the radicand? The greatest perfect cube that divides each of the factors in the radicand is $5^3$ for the factor $250$, $(x^3)^4$ for the factor $x^{12}$, $(y^3)^2$ for the factor $y^7$, and $(z^3)^6$ for the factor $z^{20}$.
• How do we simplify the expression using the properties of radicals? We simplify the expression by applying the property that $\sqrt[3]{a^3} = a$ to simplify the expression.
• What is the final simplified expression? The final simplified expression is $-7 \cdot 2 \cdot 5 \cdot x^4 \cdot y^2 \cdot z^6 \cdot \sqrt[3]{y \cdot z^2}$.

Step-by-Step Solution

1. Break down the radicand into its prime factors.
2. Factor out perfect cubes from the radicand.
3. Simplify the expression by combining like terms.
4. Apply the properties of radicals to simplify the expression.
5. Simplify the expression further by applying the properties of radicals.
6. The final simplified expression is $-7 \cdot 2 \cdot 5 \cdot x^4 \cdot y^2 \cdot z^6 \cdot \sqrt[3]{y \cdot z^2}$.

Example Use Case

Suppose we want to simplify the expression $-7 \sqrt[3]{250 x^{12} y^7 z^{20}}$ using the properties of radicals. We can follow the steps outlined above to simplify the expression.

Code Solution

import math
def simplify_expression():
# Define the radicand
radicand = 250 * (x12) * (y7) * (z**20)
# Factor out perfect cubes from the radicand
perfect_cubes = 2 * 5**3 * (x**3)**4 * (y**3)**2 * (z**3)**6

# Simplify the expression by combining like terms
simplified_expression = 2**2 * 5**3 * (x**3)**4 * (y**3)**2 * (z**3)**6 * y * z**2

# Apply the properties of radicals to simplify the expression
simplified_expression = -7 * 2 * 5 * x**4 * y**2 * z**6 * math.pow(y * z**2, 1/3)

return simplified_expression

x = 1
y = 1
z = 1

simplified_expression = simplify_expression()
print(simplified_expression)

Conclusion

In this article, we simplified the expression $-7 \sqrt[3]{250 x^{12} y^7 z^{20}}$ by factoring out perfect cubes and applying the properties of radicals. We broke down the radicand into its prime factors, factored out perfect cubes, and simplified the expression further by applying the properties of radicals. The final simplified expression is $-7 \cdot 2 \cdot 5 \cdot x^4 \cdot y^2 \cdot z^6 \cdot \sqrt[3]{y \cdot z^2}$.

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Frequently Asked Questions

Q: What is the greatest perfect cube that divides each of the factors in the radicand?

A: The greatest perfect cube that divides each of the factors in the radicand is $5^3$ for the factor $250$, $(x^3)^4$ for the factor $x^{12}$, $(y^3)^2$ for the factor $y^7$, and $(z^3)^6$ for the factor $z^{20}$.

Q: How do we simplify the expression using the properties of radicals?

A: We simplify the expression by applying the property that $\sqrt[3]{a^3} = a$ to simplify the expression.

Q: What is the final simplified expression?

A: The final simplified expression is $-7 \cdot 2 \cdot 5 \cdot x^4 \cdot y^2 \cdot z^6 \cdot \sqrt[3]{y \cdot z^2}$.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by applying the properties of radicals.

Q: How do we apply the properties of radicals to simplify the expression?

A: We apply the properties of radicals by using the property that $\sqrt[3]{a^3} = a$ to simplify the expression.

Q: What is the significance of the property $\sqrt[3]{a^3} = a$ in simplifying the expression?

A: The property $\sqrt[3]{a^3} = a$ is significant in simplifying the expression because it allows us to simplify the expression by removing the cube root.

Q: Can we use the property $\sqrt[3]{a^3} = a$ to simplify other expressions?

A: Yes, we can use the property $\sqrt[3]{a^3} = a$ to simplify other expressions that involve cube roots.

Q: How do we determine if an expression can be simplified using the property $\sqrt[3]{a^3} = a$?

A: We determine if an expression can be simplified using the property $\sqrt[3]{a^3} = a$ by checking if the radicand is a perfect cube.

Q: What is a perfect cube?

A: A perfect cube is a number that can be expressed as the cube of an integer, such as $1^3 = 1$, $2^3 = 8$, or $3^3 = 27$.

Q: Can we simplify the expression $-7 \sqrt[3]{250 x^{12} y^7 z^{20}}$ using other methods?

A: Yes, we can simplify the expression $-7 \sqrt[3]{250 x^{12} y^7 z^{20}}$ using other methods, such as factoring out perfect squares or using the properties of exponents.

Q: What are some other methods for simplifying expressions involving radicals?

A: Some other methods for simplifying expressions involving radicals include factoring out perfect squares, using the properties of exponents, and applying the properties of radicals.

Q: Can we use the properties of radicals to simplify expressions involving other types of roots?

A: Yes, we can use the properties of radicals to simplify expressions involving other types of roots, such as square roots or fourth roots.

Q: How do we apply the properties of radicals to simplify expressions involving other types of roots?

A: We apply the properties of radicals to simplify expressions involving other types of roots by using the corresponding properties of the root, such as the property that $\sqrt{a^2} = a$ for square roots.

Q: What are some common properties of radicals that we can use to simplify expressions?

A: Some common properties of radicals that we can use to simplify expressions include the property that $\sqrt[3]{a^3} = a$, the property that $\sqrt{a^2} = a$, and the property that $\sqrt[4]{a^4} = a$.

Q: Can we use the properties of radicals to simplify expressions involving complex numbers?

A: Yes, we can use the properties of radicals to simplify expressions involving complex numbers.

Q: How do we apply the properties of radicals to simplify expressions involving complex numbers?

A: We apply the properties of radicals to simplify expressions involving complex numbers by using the corresponding properties of the root, such as the property that $\sqrt{a^2} = a$ for square roots.

Q: What are some common properties of radicals that we can use to simplify expressions involving complex numbers?

A: Some common properties of radicals that we can use to simplify expressions involving complex numbers include the property that $\sqrt[3]{a^3} = a$, the property that $\sqrt{a^2} = a$, and the property that $\sqrt[4]{a^4} = a$.

Conclusion

In this article, we answered some frequently asked questions about simplifying the expression $-7 \sqrt[3]{250 x^{12} y^7 z^{20}}$. We discussed the properties of radicals and how to apply them to simplify expressions involving cube roots. We also discussed other methods for simplifying expressions involving radicals and how to apply the properties of radicals to simplify expressions involving complex numbers.