Answer: For The Expression 49 Z 10 \sqrt{49 Z^{10}} 49 Z 10 ​ To Simplify So That The Root Disappears, The Exponent On The Variable Must Be Even. Show The Full Solution.In Question 1, The Expression Simplifies To 7 Z 5 7 Z^{5} 7 Z 5 . For This To Happen, What

Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will delve into the world of radical expressions and explore the conditions under which they can be simplified to remove the radical sign. We will also examine a specific example, Question 1, and demonstrate how to simplify the expression $\sqrt{49 z^{10}}$ to $7 z^{5}$.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a radical sign. The most common type of radical expression is the square root, denoted by $\sqrt{x}$. However, other roots, such as cube roots, fourth roots, and so on, are also possible. In this article, we will focus on square roots, but the principles we discuss can be applied to other types of roots as well.

Simplifying Radical Expressions

To simplify a radical expression, we need to follow a few simple rules:

1. Separate the radicand: The radicand is the expression inside the radical sign. We can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, we can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once we have simplified each factor, we can multiply them together to get the final simplified expression.

Example: Simplifying $\sqrt{49 z^{10}}$

Now, let's apply these rules to the expression $\sqrt{49 z^{10}}$. To simplify this expression, we need to separate the radicand into two factors: $49$ and $z^{10}$.

$\sqrt{49 z^{10}} = \sqrt{49} \cdot \sqrt{z^{10}}$

Next, we need to simplify each factor. The square root of $49$ is $7$, and the square root of $z^{10}$ is $z^{5}$.

$\sqrt{49} = 7$
$\sqrt{z^{10}} = z^{5}$

Finally, we multiply the simplified factors together to get the final simplified expression.

$\sqrt{49 z^{10}} = 7 \cdot z^{5} = 7 z^{5}$

Conclusion

In this article, we have explored the conditions under which radical expressions can be simplified to remove the radical sign. We have also demonstrated how to simplify the expression $\sqrt{49 z^{10}}$ to $7 z^{5}$. By following the simple rules outlined above, you can simplify any radical expression and remove the radical sign.

Key Takeaways

• To simplify a radical expression, separate the radicand into two or more factors, as long as each factor is a perfect square.
• Simplify each factor by taking the square root of the factor.
• Multiply the simplified factors together to get the final simplified expression.
• The exponent on the variable must be even for the root to disappear.

Further Reading

If you want to learn more about radical expressions and how to simplify them, here are some additional resources:

• Khan Academy: Radical Expressions and Equations
• Mathway: Simplifying Radical Expressions
• Wolfram Alpha: Simplifying Radical Expressions

Practice Problems

Try simplifying the following radical expressions:

• $\sqrt{16 x^{12}}$
• $\sqrt{25 y^{20}}$
• $\sqrt{36 z^{24}}$

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a radical sign. The most common type of radical expression is the square root, denoted by $\sqrt{x}$. However, other roots, such as cube roots, fourth roots, and so on, are also possible.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: What is a perfect square?

A: A perfect square is a number or expression that can be expressed as the square of an integer. For example, $4$ is a perfect square because it can be expressed as $2^2$. Similarly, $9$ is a perfect square because it can be expressed as $3^2$.

Q: How do I know if a factor is a perfect square?

A: To determine if a factor is a perfect square, you can try to express it as the square of an integer. If you can do so, then the factor is a perfect square.

Q: Can I simplify a radical expression if the exponent on the variable is odd?

A: No, you cannot simplify a radical expression if the exponent on the variable is odd. The exponent on the variable must be even for the root to disappear.

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

A: A radical expression is a mathematical expression that contains a root or a radical sign, while an exponential expression is a mathematical expression that contains an exponent. For example, $\sqrt{x}$ is a radical expression, while $x^2$ is an exponential expression.

Q: Can I simplify a radical expression with a negative exponent?

A: No, you cannot simplify a radical expression with a negative exponent. The exponent on the variable must be positive for the root to disappear.

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

A: To simplify a radical expression with a variable in the radicand, you need to follow the same steps as before:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: Can I simplify a radical expression with a coefficient?

A: Yes, you can simplify a radical expression with a coefficient. The coefficient is a number that is multiplied by the radicand. You can simplify the coefficient by taking the square root of the coefficient.

Q: How do I simplify a radical expression with a coefficient and a variable in the radicand?

A: To simplify a radical expression with a coefficient and a variable in the radicand, you need to follow the same steps as before:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: Can I simplify a radical expression with a fraction in the radicand?

A: Yes, you can simplify a radical expression with a fraction in the radicand. You can simplify the fraction by taking the square root of the numerator and the denominator separately.

Q: How do I simplify a radical expression with a fraction in the radicand?

A: To simplify a radical expression with a fraction in the radicand, you need to follow these steps:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: Can I simplify a radical expression with a negative number in the radicand?

A: No, you cannot simplify a radical expression with a negative number in the radicand. The radicand must be a positive number for the root to disappear.

Q: How do I simplify a radical expression with a negative number in the radicand?

A: To simplify a radical expression with a negative number in the radicand, you need to follow these steps:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: Can I simplify a radical expression with a complex number in the radicand?

A: No, you cannot simplify a radical expression with a complex number in the radicand. The radicand must be a real number for the root to disappear.

Q: How do I simplify a radical expression with a complex number in the radicand?

A: To simplify a radical expression with a complex number in the radicand, you need to follow these steps:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: Can I simplify a radical expression with a variable in the denominator?

A: No, you cannot simplify a radical expression with a variable in the denominator. The denominator must be a positive number for the root to disappear.

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

A: To simplify a radical expression with a variable in the denominator, you need to follow these steps:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: Can I simplify a radical expression with a fraction in the denominator?

A: Yes, you can simplify a radical expression with a fraction in the denominator. You can simplify the fraction by taking the square root of the numerator and the denominator separately.

Q: How do I simplify a radical expression with a fraction in the denominator?

A: To simplify a radical expression with a fraction in the denominator, you need to follow these steps:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by taking the square root of the factor.
3. Multiply the simplified factors: Once you have simplified each factor, you can multiply them together to get the final simplified expression.

Q: Can I simplify a radical expression with a negative exponent in the denominator?

A: No, you cannot simplify a radical expression with a negative exponent in the denominator. The exponent on the variable must be positive for the root to disappear.

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

A: To simplify a radical expression with a negative exponent in the denominator, you need to follow these steps:

1. Separate the radicand: The radicand is the expression inside the radical sign. You can separate the radicand into two or more factors, as long as each factor is a perfect square.
2. Simplify each factor: If a factor is a perfect square, you can simplify it by