Enter The Correct Answer In The Box.If X \textgreater 0 X \ \textgreater \ 0 X \textgreater 0 , What Is The Difference Of 6 X 14 X − 5 56 X 3 6x\sqrt{14x} - 5\sqrt{56x^3} 6 X 14 X − 5 56 X 3 In Simplest Radical Form?

Mar 12, 2025 by ADMIN 223 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

Understanding the Problem

When dealing with radical expressions, it's essential to simplify them to their most basic form. In this problem, we're given the expression $6x\sqrt{14x} - 5\sqrt{56x^3}$ and asked to find the difference in simplest radical form. To start, let's break down the given expression and identify the key components.

Breaking Down the Expression

The given expression consists of two terms: $6x\sqrt{14x}$ and $-5\sqrt{56x^3}$ . To simplify these terms, we need to focus on the radicals and identify any common factors.

Simplifying the First Term

The first term is $6x\sqrt{14x}$ . We can start by simplifying the radical $\sqrt{14x}$ . Since $14x$ can be expressed as $2 \cdot 7 \cdot x$ , we can rewrite the radical as $\sqrt{2 \cdot 7 \cdot x}$ . This can be further simplified to $\sqrt{2} \cdot \sqrt{7} \cdot \sqrt{x}$ .

Simplifying the Second Term

The second term is $-5\sqrt{56x^3}$ . We can start by simplifying the radical $\sqrt{56x^3}$ . Since $56x^3$ can be expressed as $2^3 \cdot 7 \cdot x^3$ , we can rewrite the radical as $\sqrt{2^3 \cdot 7 \cdot x^3}$ . This can be further simplified to $2 \cdot \sqrt{7} \cdot \sqrt{x^3}$ .

Simplifying the Radical in the Second Term

Now that we have simplified the radical in the second term, we can further simplify $\sqrt{x^3}$ . Since $x^3$ can be expressed as $x^2 \cdot x$ , we can rewrite the radical as $\sqrt{x^2} \cdot \sqrt{x}$ . This simplifies to $x\sqrt{x}$ .

Combining the Simplified Terms

Now that we have simplified both terms, we can combine them to find the difference. The first term is $6x\sqrt{14x}$ , which simplifies to $6x\sqrt{2} \cdot \sqrt{7} \cdot \sqrt{x}$ . The second term is $-5\sqrt{56x^3}$ , which simplifies to $-5 \cdot 2 \cdot \sqrt{7} \cdot x\sqrt{x}$ .

Simplifying the Difference

To simplify the difference, we can combine like terms. The first term is $6x\sqrt{2} \cdot \sqrt{7} \cdot \sqrt{x}$ , and the second term is $-10\sqrt{7} \cdot x\sqrt{x}$ . We can rewrite the second term as $-10\sqrt{7} \cdot x^2\sqrt{x}$ .

Factoring Out Common Terms

Now that we have combined the terms, we can factor out common terms. The first term is $6x\sqrt{2} \cdot \sqrt{7} \cdot \sqrt{x}$ , and the second term is $-10\sqrt{7} \cdot x^2\sqrt{x}$ . We can factor out $\sqrt{7}$ from both terms.

Simplifying the Difference

After factoring out $\sqrt{7}$ , we are left with $6x\sqrt{2} \cdot \sqrt{x} - 10x^2\sqrt{x}$ . We can rewrite the first term as $6x\sqrt{2x}$ and the second term as $-10x^2\sqrt{x}$ .

Combining Like Terms

Now that we have rewritten the terms, we can combine like terms. The first term is $6x\sqrt{2x}$ , and the second term is $-10x^2\sqrt{x}$ . We can rewrite the second term as $-10x^2\sqrt{x}$ .

Simplifying the Difference

To simplify the difference, we can combine like terms. The first term is $6x\sqrt{2x}$ , and the second term is $-10x^2\sqrt{x}$ . We can rewrite the second term as $-10x^2\sqrt{x}$ .

Factoring Out Common Terms

Now that we have combined the terms, we can factor out common terms. The first term is $6x\sqrt{2x}$ , and the second term is $-10x^2\sqrt{x}$ . We can factor out $x\sqrt{x}$ from both terms.

Simplifying the Difference

After factoring out $x\sqrt{x}$ , we are left with $6\sqrt{2x} - 10x\sqrt{x}$ . We can rewrite the first term as $6\sqrt{2x}$ and the second term as $-10x\sqrt{x}$ .

Combining Like Terms

Now that we have rewritten the terms, we can combine like terms. The first term is $6\sqrt{2x}$ , and the second term is $-10x\sqrt{x}$ . We can rewrite the second term as $-10x\sqrt{x}$ .

Simplifying the Difference

To simplify the difference, we can combine like terms. The first term is $6\sqrt{2x}$ , and the second term is $-10x\sqrt{x}$ . We can rewrite the second term as $-10x\sqrt{x}$ .

Simplifying the Difference

The final simplified form of the difference is $6\sqrt{2x} - 10x\sqrt{x}$ .

Conclusion

In this problem, we were given the expression $6x\sqrt{14x} - 5\sqrt{56x^3}$ and asked to find the difference in simplest radical form. We simplified the radicals and combined like terms to arrive at the final answer. The key to simplifying radical expressions is to identify common factors and factor them out. By following these steps, we can simplify even the most complex radical expressions.

Final Answer

The final answer is: $\boxed{6\sqrt{2x} - 10x\sqrt{x}}$

Understanding Radical Expressions

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we'll 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 other root of a number. It's denoted by the symbol √ and is used to represent the value of a number that, when multiplied by itself, gives the original number.

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 number inside the radical. You can then rewrite the radical expression as the product of the perfect square and the remaining number.

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

A: A perfect square is a number that can be expressed as the product of an integer and itself, such as 4 (2 × 2) or 9 (3 × 3). A perfect cube is a number that can be expressed as the product of an integer and itself twice, such as 8 (2 × 2 × 2) or 27 (3 × 3 × 3).

Q: How do I simplify a radical expression with a perfect cube?

A: To simplify a radical expression with a perfect cube, you need to identify the largest perfect cube that divides the number inside the radical. You can then rewrite the radical expression as the product of the perfect cube and the remaining number.

Q: What is the difference between a rational and irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, such as 3/4 or 22/7. An irrational number is a number that cannot be expressed as the ratio of two integers, such as the square root of 2 or the value of pi.

Q: How do I simplify a radical expression with a rational number?

A: To simplify a radical expression with a rational number, you need to multiply the rational number by the radical expression. This will eliminate the radical and leave you with a rational number.

Q: What is the difference between a real and imaginary number?

A: A real number is a number that can be expressed as a rational or irrational number, such as 3 or the square root of 2. An imaginary number is a number that cannot be expressed as a real number, such as the square root of -1.

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

A: To simplify a radical expression with an imaginary number, you need to multiply the imaginary number by the radical expression. This will eliminate the radical and leave you with an imaginary number.

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

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

Not identifying the largest perfect square or perfect cube that divides the number inside the radical
Not rewriting the radical expression as the product of the perfect square or perfect cube and the remaining number
Not multiplying the rational or imaginary number by the radical expression
Not eliminating the radical and leaving the expression in its original form

Conclusion

Simplifying radical expressions is a crucial skill to master in mathematics. By understanding the concepts of perfect squares, perfect cubes, rational and irrational numbers, and real and imaginary numbers, you can simplify even the most complex radical expressions. Remember to identify the largest perfect square or perfect cube that divides the number inside the radical, rewrite the radical expression as the product of the perfect square or perfect cube and the remaining number, and multiply the rational or imaginary number by the radical expression. With practice and patience, you'll become a pro at simplifying radical expressions in no time!

Final Tips

Practice simplifying radical expressions regularly to build your skills and confidence.
Use online resources and math tools to help you simplify radical expressions.
Don't be afraid to ask for help if you're struggling with a particular problem or concept.
Remember to check your work and double-check your answers to ensure accuracy.

Common Radical Expressions

Here are some common radical expressions that you may encounter:

√x
√x^2
√x^3
√(x + y)
√(x - y)
√(x^2 + y^2)
√(x^2 - y^2)

Simplifying Radical Expressions: A Summary

To simplify a radical expression, you need to:

Identify the largest perfect square or perfect cube that divides the number inside the radical.
Rewrite the radical expression as the product of the perfect square or perfect cube and the remaining number.
Multiply the rational or imaginary number by the radical expression.
Eliminate the radical and leave the expression in its simplified form.

By following these steps, you can simplify even the most complex radical expressions and become a pro at math in no time!