Simplify The Expression Using The Properties Of Rational Exponents:1. Start With The Given Expression: $ \sqrt[3]{875 X^5 Y^9} }$2. Rewrite The Expression With Rational Exponents $[ (125 \cdot 7)^{\frac{2 {3}} \cdot

Introduction

Rational exponents are a powerful tool in algebra, allowing us to simplify complex expressions and solve equations with ease. In this article, we will explore how to simplify the expression $\sqrt[3]{875 x^5 y^9}$ using the properties of rational exponents.

Step 1: Factor the Radicand

The first step in simplifying the expression is to factor the radicand, which is the number inside the cube root. We can factor 875 as $125 \cdot 7$, and $x^5$ as $x^3 \cdot x^2$, and $y^9$ as $y^6 \cdot y^3$.

\sqrt[3]{875 x^5 y^9} = \sqrt[3]{125 \cdot 7 \cdot x^3 \cdot x^2 \cdot y^6 \cdot y^3}

Step 2: Rewrite the Expression with Rational Exponents

Now that we have factored the radicand, we can rewrite the expression with rational exponents. We know that $\sqrt[3]{a} = a^{\frac{1}{3}}$, so we can rewrite the expression as:

(125 \cdot 7)^{\frac{1}{3}} \cdot (x^3)^{\frac{1}{3}} \cdot (x^2)^{\frac{1}{3}} \cdot (y^6)^{\frac{1}{3}} \cdot (y^3)^{\frac{1}{3}}

Step 3: Simplify the Expression

Now that we have rewritten the expression with rational exponents, we can simplify it by combining the exponents. We know that $a^{\frac{1}{3}} \cdot a^{\frac{1}{3}} = a^{\frac{2}{3}}$, so we can simplify the expression as:

(125 \cdot 7)^{\frac{2}{3}} \cdot x \cdot x^{\frac{2}{3}} \cdot y^2 \cdot y^{\frac{2}{3}}

Step 4: Final Simplification

Finally, we can simplify the expression by combining the exponents and multiplying the coefficients. We know that $a^{\frac{2}{3}} \cdot a^{\frac{2}{3}} = a^{\frac{4}{3}}$, so we can simplify the expression as:

(125 \cdot 7)^{\frac{2}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{11}{3}}

Conclusion

In this article, we have simplified the expression $\sqrt[3]{875 x^5 y^9}$ using the properties of rational exponents. We have factored the radicand, rewritten the expression with rational exponents, simplified the expression, and finally simplified the expression to its final form. This example demonstrates the power of rational exponents in simplifying complex expressions and solving equations.

Properties of Rational Exponents

Rational exponents have several important properties that make them useful in algebra. Some of the key properties of rational exponents include:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Power of a Product: $(ab)^m = a^m \cdot b^m$
• Power of a Quotient: $(\frac{a}{b})^m = \frac{a^m}{b^m}$

Examples of Rational Exponents

Rational exponents can be used to simplify a wide range of expressions. Here are a few examples:

• Simplifying a Cube Root: $\sqrt[3]{8} = 2$
• Simplifying a Square Root: $\sqrt{16} = 4$
• Simplifying a Rational Exponent: $(2^3)^{\frac{1}{2}} = 2^{\frac{3}{2}} = 2 \cdot \sqrt{2}$

Applications of Rational Exponents

Rational exponents have many applications in mathematics and science. Some of the key applications of rational exponents include:

• Simplifying Complex Expressions: Rational exponents can be used to simplify complex expressions and solve equations.
• Modeling Real-World Problems: Rational exponents can be used to model real-world problems, such as population growth and chemical reactions.
• Solving Equations: Rational exponents can be used to solve equations, such as quadratic equations and polynomial equations.

Conclusion

Q: What are rational exponents?

A: Rational exponents are a way of expressing roots of numbers using fractions. For example, $\sqrt[3]{8}$ can be written as $8^{\frac{1}{3}}$.

Q: How do I simplify an expression with rational exponents?

A: To simplify an expression with rational exponents, you can use the following steps:

1. Factor the radicand (the number inside the root).
2. Rewrite the expression with rational exponents.
3. Simplify the expression by combining the exponents.
4. Final simplification by combining the exponents and multiplying the coefficients.

Q: What are some common properties of rational exponents?

A: Some common properties of rational exponents include:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Power of a Product: $(ab)^m = a^m \cdot b^m$
• Power of a Quotient: $(\frac{a}{b})^m = \frac{a^m}{b^m}$

Q: How do I apply rational exponents to solve equations?

A: To apply rational exponents to solve equations, you can use the following steps:

1. Rewrite the equation with rational exponents.
2. Simplify the equation by combining the exponents.
3. Solve the equation using algebraic methods.

Q: What are some real-world applications of rational exponents?

A: Some real-world applications of rational exponents include:

• Modeling population growth: Rational exponents can be used to model population growth and decline.
• Chemical reactions: Rational exponents can be used to model chemical reactions and predict the outcome of a reaction.
• Engineering: Rational exponents can be used to design and optimize systems, such as bridges and buildings.

Q: How do I choose the right rational exponent to use in a problem?

A: To choose the right rational exponent to use in a problem, you need to consider the following factors:

• The type of problem: Different types of problems require different rational exponents.
• The level of difficulty: More complex problems may require more complex rational exponents.
• The context of the problem: The context of the problem may require a specific rational exponent.

Q: What are some common mistakes to avoid when working with rational exponents?

A: Some common mistakes to avoid when working with rational exponents include:

• Not simplifying the expression: Failing to simplify the expression can lead to incorrect answers.
• Not using the correct rational exponent: Using the wrong rational exponent can lead to incorrect answers.
• Not considering the context of the problem: Failing to consider the context of the problem can lead to incorrect answers.

Q: How do I practice using rational exponents?

A: To practice using rational exponents, you can try the following:

• Work through examples: Work through examples of rational exponents to practice simplifying and solving equations.
• Practice with different types of problems: Practice with different types of problems, such as population growth and chemical reactions.
• Use online resources: Use online resources, such as video tutorials and practice problems, to help you learn and practice using rational exponents.

Conclusion

In conclusion, rational exponents are a powerful tool in algebra, allowing us to simplify complex expressions and solve equations with ease. By understanding the properties of rational exponents and how to apply them, we can solve a wide range of problems in mathematics and science.