Select The Correct Answer.What Is The Simplest Form Of $9 \sqrt{81 X^{12}}$?A. $81 X^6$B. \$729 X^6$[/tex\]C. $81 X^3$D. $729 X^3$

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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 expression $9 \sqrt{81 x^{12}}$ and selecting the correct answer from the given options.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Simplifying the Expression

To simplify the expression $9 \sqrt{81 x^{12}}$, we need to follow a step-by-step approach.

Step 1: Simplify the Radicand

The radicand is the number inside the square root. In this case, the radicand is $81 x^{12}$. We can simplify this expression by finding the prime factorization of 81.

81=3481 = 3^4

So, we can rewrite the radicand as:

81x12=(34)x1281 x^{12} = (3^4) x^{12}

Step 2: Simplify the Expression Inside the Square Root

Now, we can simplify the expression inside the square root by combining the powers of 3 and x.

(34)x12=(34)(x6)2\sqrt{(3^4) x^{12}} = \sqrt{(3^4) (x^6)^2}

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression further.

(34)(x6)2=32x6\sqrt{(3^4) (x^6)^2} = 3^2 x^6

Step 3: Simplify the Expression Outside the Square Root

The expression outside the square root is 9. We can simplify this expression by rewriting it as a power of 3.

9=329 = 3^2

So, we can rewrite the original expression as:

981x12=32â‹…32x69 \sqrt{81 x^{12}} = 3^2 \cdot 3^2 x^6

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression further.

32â‹…32x6=34x63^2 \cdot 3^2 x^6 = 3^4 x^6

Step 4: Simplify the Expression Further

Now, we can simplify the expression further by evaluating the power of 3.

34=813^4 = 81

So, we can rewrite the expression as:

81x681 x^6

Conclusion

In conclusion, the simplest form of the expression $9 \sqrt{81 x^{12}}$ is $81 x^6$. This is the correct answer from the given options.

Answer

The correct answer is:

  • A. $81 x^6$

Why is this the correct answer?

This is the correct answer because we simplified the expression step-by-step, and the final simplified expression is $81 x^6$. This expression cannot be simplified further, and it is the simplest form of the original expression.

Tips and Tricks

Here are some tips and tricks to help you simplify radical expressions:

  • Use the property of exponents: When simplifying radical expressions, use the property of exponents that states $a^m \cdot a^n = a^{m+n}$.
  • Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
  • Combine powers: Combine powers of the same base by adding the exponents.
  • Evaluate powers: Evaluate powers of the base by multiplying the base by itself as many times as the exponent indicates.

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 a mathematical expression that contains a square root or a higher root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, follow these steps:

  1. Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
  2. Combine powers: Combine powers of the same base by adding the exponents.
  3. Evaluate powers: Evaluate powers of the base by multiplying the base by itself as many times as the exponent indicates.
  4. Simplify the expression: Simplify the expression by combining the simplified radicand and the powers of the base.

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

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while an exponential expression is a mathematical expression that contains a power of a number. For example, $\sqrt{16}$ is a radical expression, while $2^4$ is an exponential expression.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, follow these steps:

  1. Simplify each radical: Simplify each radical expression separately by following the steps outlined above.
  2. Combine the simplified radicals: Combine the simplified radicals by adding or subtracting the powers of the base.
  3. Simplify the expression: Simplify the expression by combining the simplified radicals and the powers of the base.

Q: What is the order of operations for simplifying radical expressions?

A: The order of operations for simplifying radical expressions is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate expressions with exponents next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

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

A: To simplify a radical expression with a variable, follow these steps:

  1. Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
  2. Combine powers: Combine powers of the same base by adding the exponents.
  3. Evaluate powers: Evaluate powers of the base by multiplying the base by itself as many times as the exponent indicates.
  4. Simplify the expression: Simplify the expression by combining the simplified radicand and the powers of the base.

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

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

  • Not simplifying the radicand: Failing to simplify the radicand can lead to incorrect simplifications.
  • Not combining powers: Failing to combine powers of the same base can lead to incorrect simplifications.
  • Not evaluating powers: Failing to evaluate powers of the base can lead to incorrect simplifications.
  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect simplifications.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By following the steps outlined in this Q&A guide, you can simplify radical expressions with ease and avoid common mistakes. Remember to simplify the radicand, combine powers, evaluate powers, and follow the order of operations to simplify radical expressions correctly.