Perform The Indicated Operation And Write The Answer In Simplest Form. Assume All Variables Are Positive. 9 W 5 − W W 3 = □ \sqrt{9 W^5} - W \sqrt{w^3} = \square 9 W 5 ​ − W W 3 ​ = □

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the given radical expression $\sqrt{9 w^5} - w \sqrt{w^3}$. We will break down the problem into manageable steps and provide a clear explanation of each step.

Step 1: Simplify the First Radical Expression

The first radical expression is $\sqrt{9 w^5}$. To simplify this expression, we need to identify the largest perfect square that divides the radicand (the number inside the square root). In this case, the largest perfect square that divides $9 w^5$ is $9 w^4$.

$\sqrt{9 w^5} = \sqrt{9 w^4 \cdot w} = \sqrt{9 w^4} \cdot \sqrt{w} = 3 w^2 \sqrt{w}$

Step 2: Simplify the Second Radical Expression

The second radical expression is $w \sqrt{w^3}$. To simplify this expression, we need to identify the largest perfect square that divides the radicand (the number inside the square root). In this case, the largest perfect square that divides $w^3$ is $w^2$.

$w \sqrt{w^3} = w \sqrt{w^2 \cdot w} = w \sqrt{w^2} \cdot \sqrt{w} = w^2 \sqrt{w}$

Step 3: Combine the Simplified Radical Expressions

Now that we have simplified both radical expressions, we can combine them to get the final result.

$\sqrt{9 w^5} - w \sqrt{w^3} = 3 w^2 \sqrt{w} - w^2 \sqrt{w}$

Step 4: Factor Out the Common Term

The two terms in the expression have a common factor of $w^2 \sqrt{w}$. We can factor this out to simplify the expression further.

$3 w^2 \sqrt{w} - w^2 \sqrt{w} = (3 - 1) w^2 \sqrt{w} = 2 w^2 \sqrt{w}$

Conclusion

In this article, we simplified the given radical expression $\sqrt{9 w^5} - w \sqrt{w^3}$ by breaking it down into manageable steps. We identified the largest perfect square that divides each radicand, simplified each radical expression, and combined them to get the final result. The final answer is $\boxed{2 w^2 \sqrt{w}}$.

Tips and Tricks

• When simplifying radical expressions, always identify the largest perfect square that divides the radicand.
• Use the properties of square roots to simplify the expression.
• Factor out common terms to simplify the expression further.

Common Mistakes to Avoid

• Failing to identify the largest perfect square that divides the radicand.
• Not using the properties of square roots to simplify the expression.
• Not factoring out common terms to simplify the expression further.

Real-World Applications

Simplifying radical expressions is a crucial skill in mathematics, and it has many real-world applications. For example, in physics, radical expressions are used to describe the motion of objects. In engineering, radical expressions are used to describe the behavior of electrical circuits. In finance, radical expressions are used to describe the growth of investments.

Practice Problems

1. Simplify the radical expression $\sqrt{16 x^7} - 2 x \sqrt{x^3}$.
2. Simplify the radical expression $\sqrt{25 y^9} - 3 y \sqrt{y^3}$.
3. Simplify the radical expression $\sqrt{36 z^5} - 4 z \sqrt{z^3}$.

Answer Key

1. $\boxed{4 x^3 \sqrt{x} - 2 x^3 \sqrt{x}} = 2 x^3 \sqrt{x}$
2. $\boxed{5 y^4 \sqrt{y} - 3 y^4 \sqrt{y}} = 2 y^4 \sqrt{y}$
3. $\boxed{6 z^2 \sqrt{z} - 4 z^2 \sqrt{z}} = 2 z^2 \sqrt{z}$
   Simplifying Radical Expressions: A Q&A Guide =====================================================

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will provide a Q&A guide to help you understand and simplify radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or a cube root. It is denoted by the symbol $\sqrt{}$ or $\sqrt[3]{}$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to identify the largest perfect square that divides the radicand (the number inside the square root). You can then use the properties of square roots to simplify the expression.

Q: What are the properties of square roots?

A: The properties of square roots are:

• $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
• $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
• $\sqrt{a^2} = a$
• $\sqrt{a^3} = a\sqrt{a}$

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to identify the largest perfect square that divides the radicand (the number inside the square root). You can then use the properties of square roots to simplify the expression.

Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is an expression that contains a square root or a cube root, while an exponential expression is an expression that contains a power or an exponent.

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you need to rewrite the expression with a positive exponent. You can then use the properties of square roots to simplify the expression.

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

A: Some common mistakes to avoid when simplifying radical expressions are:

• Failing to identify the largest perfect square that divides the radicand.
• Not using the properties of square roots to simplify the expression.
• Not factoring out common terms to simplify the expression further.

Q: How do I check my work when simplifying radical expressions?

A: To check your work when simplifying radical expressions, you can:

• Plug in values for the variables to see if the expression simplifies correctly.
• Use a calculator to check the expression.
• Check the expression against a known solution.

Q: What are some real-world applications of simplifying radical expressions?

A: Some real-world applications of simplifying radical expressions are:

• Physics: Radical expressions are used to describe the motion of objects.
• Engineering: Radical expressions are used to describe the behavior of electrical circuits.
• Finance: Radical expressions are used to describe the growth of investments.

Q: How do I practice simplifying radical expressions?

A: To practice simplifying radical expressions, you can:

• Work through practice problems in a textbook or online resource.
• Use a calculator to check your work.
• Create your own practice problems.

Conclusion

Simplifying radical expressions is a crucial skill in mathematics, and it has many real-world applications. By understanding the properties of square roots and practicing simplifying radical expressions, you can become proficient in this skill. Remember to check your work and use a calculator to verify your answers.

Tips and Tricks

• Always identify the largest perfect square that divides the radicand.
• Use the properties of square roots to simplify the expression.
• Factor out common terms to simplify the expression further.

Common Mistakes to Avoid

• Failing to identify the largest perfect square that divides the radicand.
• Not using the properties of square roots to simplify the expression.
• Not factoring out common terms to simplify the expression further.

Real-World Applications

• Physics: Radical expressions are used to describe the motion of objects.
• Engineering: Radical expressions are used to describe the behavior of electrical circuits.
• Finance: Radical expressions are used to describe the growth of investments.

Practice Problems

1. Simplify the radical expression $\sqrt{16 x^7} - 2 x \sqrt{x^3}$.
2. Simplify the radical expression $\sqrt{25 y^9} - 3 y \sqrt{y^3}$.
3. Simplify the radical expression $\sqrt{36 z^5} - 4 z \sqrt{z^3}$.

Answer Key

1. $\boxed{4 x^3 \sqrt{x} - 2 x^3 \sqrt{x}} = 2 x^3 \sqrt{x}$
2. $\boxed{5 y^4 \sqrt{y} - 3 y^4 \sqrt{y}} = 2 y^4 \sqrt{y}$
3. $\boxed{6 z^2 \sqrt{z} - 4 z^2 \sqrt{z}} = 2 z^2 \sqrt{z}$