Simplify The Expression:$\[ \sqrt[3]{\frac{10 Y^5}{20 X^6}} \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including exponents, radicals, and fractions. In this article, we will focus on simplifying the given expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$. We will break down the expression into smaller parts, simplify each part, and then combine them to obtain the final simplified expression.

Understanding the Expression

The given expression is a radical expression, which involves a cube root. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In this case, we have $\sqrt[3]{\frac{10 y^5}{20 x^6}}$, which means we need to find the cube root of the fraction $\frac{10 y^5}{20 x^6}$.

Simplifying the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 10 and 20 is 10. We can divide both the numerator and the denominator by 10 to simplify the fraction:

$$\frac{10 y^5}{20 x^6} = \frac{1 y^5}{2 x^6}$$

Simplifying the Exponents

Now that we have simplified the fraction, we can focus on simplifying the exponents. In this case, we have $y^5$ and $x^6$. We can simplify the exponents by dividing the exponent of $y$ by 3, since we are taking the cube root:

$$y^5 = y^{5/3}$$

Similarly, we can simplify the exponent of $x$ by dividing the exponent of $x$ by 3:

$$x^6 = x^{6/3} = x^2$$

Simplifying the Radical Expression

Now that we have simplified the fraction and the exponents, we can simplify the radical expression. We can rewrite the expression as:

$$\sqrt[3]{\frac{1 y^5}{2 x^6}} = \sqrt[3]{\frac{1 y^{5/3}}{2 x^2}}$$

Final Simplification

To simplify the expression further, we can take the cube root of the numerator and the denominator separately:

$$\sqrt[3]{\frac{1 y^{5/3}}{2 x^2}} = \frac{\sqrt[3]{1 y^{5/3}}}{\sqrt[3]{2 x^2}}$$

Since $\sqrt[3]{1} = 1$, we can simplify the expression as:

$$\frac{\sqrt[3]{1 y^{5/3}}}{\sqrt[3]{2 x^2}} = \frac{y^{5/9}}{\sqrt[3]{2} x^{2/3}}$$

Conclusion

In this article, we simplified the given expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$ by breaking it down into smaller parts, simplifying each part, and then combining them to obtain the final simplified expression. We used various mathematical concepts, including exponents, radicals, and fractions, to simplify the expression. The final simplified expression is $\frac{y^{5/9}}{\sqrt[3]{2} x^{2/3}}$.

Frequently Asked Questions

• What is the cube root of a number? The cube root of a number is a value that, when multiplied by itself twice, gives the original number.
• How do you simplify a radical expression? To simplify a radical expression, you need to simplify the fraction and the exponents, and then take the cube root of the numerator and the denominator separately.
• What is the greatest common divisor (GCD) of two numbers? The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

Step-by-Step Solution

1. Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
2. Simplify the exponents by dividing the exponent of $y$ by 3 and the exponent of $x$ by 3.
3. Simplify the radical expression by taking the cube root of the numerator and the denominator separately.
4. Combine the simplified parts to obtain the final simplified expression.

Example Problems

• Simplify the expression $\sqrt[3]{\frac{15 x^7}{30 y^8}}$.
• Simplify the expression $\sqrt[3]{\frac{20 z^9}{40 w^{10}}}$.

Real-World Applications

Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and analyze complex systems. In economics, we use algebraic expressions to model and analyze economic systems.

Conclusion

In this article, we simplified the given expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$ by breaking it down into smaller parts, simplifying each part, and then combining them to obtain the final simplified expression. We used various mathematical concepts, including exponents, radicals, and fractions, to simplify the expression. The final simplified expression is $\frac{y^{5/9}}{\sqrt[3]{2} x^{2/3}}$. We also discussed the importance of simplifying algebraic expressions in real-world applications.

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Introduction

In our previous article, we simplified the given expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$ by breaking it down into smaller parts, simplifying each part, and then combining them to obtain the final simplified expression. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions, including the given expression.

Q&A

Q: What is the cube root of a number?

A: The cube root of a number is a value that, when multiplied by itself twice, gives the original number.

Q: How do you simplify a radical expression?

A: To simplify a radical expression, you need to simplify the fraction and the exponents, and then take the cube root of the numerator and the denominator separately.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do you simplify the expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$?

A: To simplify the expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$, you need to simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator, simplify the exponents by dividing the exponent of $y$ by 3 and the exponent of $x$ by 3, and then take the cube root of the numerator and the denominator separately.

Q: What is the final simplified expression of $\sqrt[3]{\frac{10 y^5}{20 x^6}}$?

A: The final simplified expression of $\sqrt[3]{\frac{10 y^5}{20 x^6}}$ is $\frac{y^{5/9}}{\sqrt[3]{2} x^{2/3}}$.

Q: How do you apply the concept of simplifying algebraic expressions in real-world applications?

A: The concept of simplifying algebraic expressions is applied in various real-world applications, including physics, engineering, and economics. In physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and analyze complex systems. In economics, we use algebraic expressions to model and analyze economic systems.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Not simplifying the fraction and the exponents before taking the cube root.
• Not finding the greatest common divisor (GCD) of the numerator and the denominator.
• Not dividing the exponent of $y$ by 3 and the exponent of $x$ by 3.
• Not taking the cube root of the numerator and the denominator separately.

Conclusion

In this article, we answered some frequently asked questions related to simplifying algebraic expressions, including the given expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$. We discussed the concept of simplifying algebraic expressions, including the greatest common divisor (GCD) of two numbers, and the importance of simplifying algebraic expressions in real-world applications. We also highlighted some common mistakes to avoid when simplifying algebraic expressions.

Frequently Asked Questions

• What is the cube root of a number?
• How do you simplify a radical expression?
• What is the greatest common divisor (GCD) of two numbers?
• How do you simplify the expression $\sqrt[3]{\frac{10 y^5}{20 x^6}}$?
• What is the final simplified expression of $\sqrt[3]{\frac{10 y^5}{20 x^6}}$?
• How do you apply the concept of simplifying algebraic expressions in real-world applications?
• What are some common mistakes to avoid when simplifying algebraic expressions?

Step-by-Step Solution

1. Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.
2. Simplify the exponents by dividing the exponent of $y$ by 3 and the exponent of $x$ by 3.
3. Take the cube root of the numerator and the denominator separately.
4. Combine the simplified parts to obtain the final simplified expression.

Example Problems

• Simplify the expression $\sqrt[3]{\frac{15 x^7}{30 y^8}}$.
• Simplify the expression $\sqrt[3]{\frac{20 z^9}{40 w^{10}}}$.

Real-World Applications

• Simplifying algebraic expressions is a crucial skill in mathematics, and it has many real-world applications.
• In physics, we use algebraic expressions to describe the motion of objects.
• In engineering, we use algebraic expressions to design and analyze complex systems.
• In economics, we use algebraic expressions to model and analyze economic systems.