Understanding Rational Exponents

Oct 13, 2025 by miodeya.com 111 views

Introduction

Rational exponents are a powerful tool in algebra that allows us to simplify complex expressions involving roots and powers. In this article, we will explore the properties of rational exponents and apply them to simplify the given expression: $\sqrt[3]{875 x^5 y^9}$ . We will examine two possible simplification steps and determine the correct order of operations.

Understanding Rational Exponents

Rational exponents are a way of expressing roots and powers using fractions. For example, the cube root of a number can be expressed as $\sqrt[3]{x} = x^{\frac{1}{3}}$ . Similarly, the square root of a number can be expressed as $\sqrt{x} = x^{\frac{1}{2}}$ . Rational exponents can be used to simplify complex expressions involving roots and powers.

Properties of Rational Exponents

There are several properties of rational exponents that we will use to simplify the given expression. These properties include:

Product of Powers: When we multiply two powers with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$ .
Power of a Power: When we raise a power to another power, we multiply their exponents. For example, $(x^a)^b = x^{ab}$ .
Root of a Power: When we take the root of a power, we divide the exponent by the index of the root. For example, $\sqrt[n]{x^a} = x^{\frac{a}{n}}$ .

Simplifying the Expression

Now that we have a good understanding of rational exponents and their properties, let's simplify the given expression: $\sqrt[3]{875 x^5 y^9}$ . We will examine two possible simplification steps and determine the correct order of operations.

Step 1: Factor the Coefficient

The first step in simplifying the expression is to factor the coefficient, 875. We can factor 875 as $125 \cdot 7$ . Therefore, we can rewrite the expression as:

$\sqrt[3]{125 \cdot 7 \cdot x^5 y^9}$

Step 2: Apply the Product of Powers Property

Now that we have factored the coefficient, we can apply the product of powers property to simplify the expression. The product of powers property states that when we multiply two powers with the same base, we add their exponents. Therefore, we can rewrite the expression as:

$\sqrt[3]{125 \cdot 7} \cdot \sqrt[3]{x^5 y^9}$

Step 3: Simplify the Cube Root of the Coefficient

The next step is to simplify the cube root of the coefficient, 125. We can simplify the cube root of 125 as:

$\sqrt[3]{125} = 5$

Therefore, we can rewrite the expression as:

$5 \cdot \sqrt[3]{7} \cdot \sqrt[3]{x^5 y^9}$

Step 4: Apply the Power of a Power Property

Now that we have simplified the cube root of the coefficient, we can apply the power of a power property to simplify the expression. The power of a power property states that when we raise a power to another power, we multiply their exponents. Therefore, we can rewrite the expression as:

$5 \cdot \sqrt[3]{7} \cdot x^{\frac{5}{3}} \cdot y^3$

Step 5: Simplify the Expression

The final step is to simplify the expression by combining the terms. We can rewrite the expression as:

$5 \cdot \sqrt[3]{7} \cdot x^{\frac{5}{3}} \cdot y^3$

Therefore, the correct order of simplification steps is:

Factor the coefficient
Apply the product of powers property
Simplify the cube root of the coefficient
Apply the power of a power property
Simplify the expression

Conclusion

In this article, we have explored the properties of rational exponents and applied them to simplify the given expression: $\sqrt[3]{875 x^5 y^9}$ . We have examined two possible simplification steps and determined the correct order of operations. The correct order of simplification steps is to factor the coefficient, apply the product of powers property, simplify the cube root of the coefficient, apply the power of a power property, and finally simplify the expression. By following these steps, we can simplify complex expressions involving roots and powers using rational exponents.

References

[1] "Rational Exponents" by Math Open Reference
[2] "Properties of Rational Exponents" by Purplemath
[3] "Simplifying Expressions with Rational Exponents" by Mathway

Discussion

Introduction

In our previous article, we explored the properties of rational exponents and applied them to simplify the expression: $\sqrt[3]{875 x^5 y^9}$ . We examined two possible simplification steps and determined the correct order of operations. In this article, we will answer some frequently asked questions about simplifying expressions with rational exponents.

Q: What is the difference between a rational exponent and a fractional exponent?

A: A rational exponent is a way of expressing roots and powers using fractions. For example, the cube root of a number can be expressed as $\sqrt[3]{x} = x^{\frac{1}{3}}$ . A fractional exponent, on the other hand, is a way of expressing a power as a fraction. For example, $x^{\frac{1}{2}}$ is equivalent to $\sqrt{x}$ .

Q: How do I simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, you can follow these steps:

Factor the coefficient
Apply the product of powers property
Simplify the cube root of the coefficient
Apply the power of a power property
Simplify the expression

Q: What is the product of powers property?

A: The product of powers property states that when we multiply two powers with the same base, we add their exponents. For example, $x^a \cdot x^b = x^{a+b}$ .

Q: What is the power of a power property?

A: The power of a power property states that when we raise a power to another power, we multiply their exponents. For example, $(x^a)^b = x^{ab}$ .

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

A: To simplify an expression with a cube root, you can follow these steps:

Factor the coefficient
Apply the product of powers property
Simplify the cube root of the coefficient
Apply the power of a power property
Simplify the expression

Q: What is the difference between a cube root and a square root?

A: A cube root is a way of expressing a number as the third root of a product. For example, $\sqrt[3]{x}$ is equivalent to $x^{\frac{1}{3}}$ . A square root, on the other hand, is a way of expressing a number as the second root of a product. For example, $\sqrt{x}$ is equivalent to $x^{\frac{1}{2}}$ .

Q: How do I simplify an expression with a rational exponent and a cube root?

A: To simplify an expression with a rational exponent and a cube root, you can follow these steps:

Factor the coefficient
Apply the product of powers property
Simplify the cube root of the coefficient
Apply the power of a power property
Simplify the expression

Conclusion

In this article, we have answered some frequently asked questions about simplifying expressions with rational exponents. We have explored the properties of rational exponents and applied them to simplify complex expressions. By following the steps outlined in this article, you can simplify expressions with rational exponents and become more confident in your math skills.

References

[1] "Rational Exponents" by Math Open Reference
[2] "Properties of Rational Exponents" by Purplemath
[3] "Simplifying Expressions with Rational Exponents" by Mathway

Discussion

What are some other examples of simplifying expressions with rational exponents? How can we apply the properties of rational exponents to simplify complex expressions? Share your thoughts and examples in the comments below!