Simplify The Following Expressions Completely:A) $\sqrt{25 P^{34}} =$ $\square$B) $\sqrt{81 M^{18}} =$ $\square$

In this article, we will simplify two given expressions completely. The first expression is $\sqrt{25 p^{34}}$ and the second expression is $\sqrt{81 m^{18}}$. We will use the properties of radicals and exponents to simplify these expressions.

Simplifying the First Expression

The first expression is $\sqrt{25 p^{34}}$. To simplify this expression, we need to find the prime factorization of the number inside the radical.

Prime Factorization of 25

The prime factorization of 25 is $5^2$. Therefore, we can rewrite the expression as:

$$\sqrt{25 p^{34}} = \sqrt{5^2 p^{34}}$$

Using the Property of Radicals

We can use the property of radicals that states $\sqrt{a^2} = a$ to simplify the expression further.

$$\sqrt{5^2 p^{34}} = 5 p^{17}$$

Therefore, the simplified form of the first expression is $5 p^{17}$.

Simplifying the Second Expression

The second expression is $\sqrt{81 m^{18}}$. To simplify this expression, we need to find the prime factorization of the number inside the radical.

Prime Factorization of 81

The prime factorization of 81 is $3^4$. Therefore, we can rewrite the expression as:

$$\sqrt{81 m^{18}} = \sqrt{3^4 m^{18}}$$

Using the Property of Radicals

We can use the property of radicals that states $\sqrt{a^2} = a$ to simplify the expression further.

$$\sqrt{3^4 m^{18}} = 3^2 m^9$$

Therefore, the simplified form of the second expression is $3^2 m^9$.

Conclusion

In this article, we simplified two given expressions completely. The first expression was $\sqrt{25 p^{34}}$ and the second expression was $\sqrt{81 m^{18}}$. We used the properties of radicals and exponents to simplify these expressions. The simplified forms of the expressions are $5 p^{17}$ and $3^2 m^9$ respectively.

Properties of Radicals and Exponents

In this section, we will discuss some of the properties of radicals and exponents that we used to simplify the expressions.

Property 1: $\sqrt{a^2} = a$

This property states that the square root of a number squared is equal to the number itself. For example, $\sqrt{4^2} = 4$.

Property 2: $\sqrt{ab} = \sqrt{a} \sqrt{b}$

This property states that the square root of a product is equal to the product of the square roots. For example, $\sqrt{12} = \sqrt{4} \sqrt{3} = 2 \sqrt{3}$.

Property 3: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

This property states that the square root of a fraction is equal to the fraction of the square roots. For example, $\sqrt{\frac{12}{16}} = \frac{\sqrt{12}}{\sqrt{16}} = \frac{2 \sqrt{3}}{4} = \frac{\sqrt{3}}{2}$.

Examples of Simplifying Expressions

In this section, we will provide some examples of simplifying expressions using the properties of radicals and exponents.

Example 1: $\sqrt{16 x^8}$

To simplify this expression, we need to find the prime factorization of the number inside the radical.

$$\sqrt{16 x^8} = \sqrt{4^2 x^8}$$

Using the property of radicals that states $\sqrt{a^2} = a$, we can simplify the expression further.

$$\sqrt{4^2 x^8} = 4 x^4$$

Therefore, the simplified form of the expression is $4 x^4$.

Example 2: $\sqrt{9 y^{12}}$

To simplify this expression, we need to find the prime factorization of the number inside the radical.

$$\sqrt{9 y^{12}} = \sqrt{3^2 y^{12}}$$

Using the property of radicals that states $\sqrt{a^2} = a$, we can simplify the expression further.

$$\sqrt{3^2 y^{12}} = 3 y^6$$

Therefore, the simplified form of the expression is $3 y^6$.

Conclusion

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In this article, we will answer some of the frequently asked questions related to simplifying expressions using radicals and exponents.

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

A: A radical is a mathematical operation that involves finding the square root of a number, while an exponent is a mathematical operation that involves raising a number to a power.

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

A: To simplify an expression with a radical, you need to find the prime factorization of the number inside the radical and then use the properties of radicals to simplify the expression.

Q: What are some common properties of radicals?

A: Some common properties of radicals include:

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

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, you need to use the properties of exponents to simplify the expression. For example, if you have the expression $x^{12}$, you can simplify it by using the property of exponents that states $x^{12} = (x^6)^2$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent is a mathematical operation that involves raising a number to a power, while a negative exponent is a mathematical operation that involves taking the reciprocal of a number raised to a power.

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

A: To simplify an expression with a negative exponent, you need to use the property of exponents that states $a^{-n} = \frac{1}{a^n}$.

Q: What are some common mistakes to avoid when simplifying expressions with radicals and exponents?

A: Some common mistakes to avoid when simplifying expressions with radicals and exponents include:

• Not finding the prime factorization of the number inside the radical
• Not using the properties of radicals and exponents correctly
• Not simplifying the expression completely

Q: How do I check my work when simplifying expressions with radicals and exponents?

A: To check your work when simplifying expressions with radicals and exponents, you need to:

• Make sure you have found the prime factorization of the number inside the radical
• Make sure you have used the properties of radicals and exponents correctly
• Make sure you have simplified the expression completely

Conclusion

In this article, we answered some of the frequently asked questions related to simplifying expressions using radicals and exponents. We also discussed some common properties of radicals and exponents, and provided some examples of simplifying expressions using these properties. By following the tips and avoiding the common mistakes, you can simplify expressions with radicals and exponents correctly and efficiently.

Additional Resources

For more information on simplifying expressions with radicals and exponents, you can refer to the following resources:

• Mathway: A online math problem solver that can help you simplify expressions with radicals and exponents.
• Khan Academy: A online learning platform that provides video lessons and practice exercises on simplifying expressions with radicals and exponents.
• Wolfram Alpha: A online calculator that can help you simplify expressions with radicals and exponents.

Practice Exercises

To practice simplifying expressions with radicals and exponents, you can try the following exercises:

• Simplify the expression $\sqrt{16 x^8}$
• Simplify the expression $\sqrt{9 y^{12}}$
• Simplify the expression $x^{12}$
• Simplify the expression $y^{-6}$

By practicing these exercises, you can improve your skills in simplifying expressions with radicals and exponents.