Which Choice Is Equivalent To The Expression Below?$\sqrt{50}-\sqrt{2}$A. $\sqrt{48}$B. $4 \sqrt{2}$C. 5D. $24 \sqrt{2}$

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Understanding the Problem

When dealing with radical expressions, it's essential to simplify them to their most basic form. This involves combining like terms, removing any unnecessary radicals, and expressing the result in the simplest possible way. In this article, we'll explore how to simplify the expression 50−2\sqrt{50}-\sqrt{2} and determine which of the given choices is equivalent to it.

Breaking Down the Expression

To simplify the expression 50−2\sqrt{50}-\sqrt{2}, we need to start by breaking down the square root of 50. We can do this by finding the largest perfect square that divides 50.

50=25×2=25×2=52\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}

Now that we've simplified the square root of 50, we can substitute this back into the original expression.

50−2=52−2\sqrt{50}-\sqrt{2} = 5\sqrt{2} - \sqrt{2}

Combining Like Terms

The next step is to combine like terms. In this case, we have two terms with the same radical, 2\sqrt{2}. We can combine these terms by subtracting the smaller term from the larger term.

52−2=(5−1)2=425\sqrt{2} - \sqrt{2} = (5 - 1)\sqrt{2} = 4\sqrt{2}

Evaluating the Choices

Now that we've simplified the expression, we can evaluate the given choices to determine which one is equivalent to it.

A. 48\sqrt{48}: This choice is not equivalent to the simplified expression. To see why, let's simplify the square root of 48.

48=16×3=16×3=43\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}

This is not equal to the simplified expression, so we can rule out choice A.

B. 424 \sqrt{2}: This choice is equivalent to the simplified expression. We can see this by comparing the two expressions.

42=424\sqrt{2} = 4\sqrt{2}

This is a perfect match, so we can conclude that choice B is the correct answer.

C. 5: This choice is not equivalent to the simplified expression. We can see why by comparing the two expressions.

5=52−25 = 5\sqrt{2} - \sqrt{2}

This is not equal to the simplified expression, so we can rule out choice C.

D. 24224 \sqrt{2}: This choice is not equivalent to the simplified expression. We can see why by comparing the two expressions.

242=24224\sqrt{2} = 24\sqrt{2}

This is not equal to the simplified expression, so we can rule out choice D.

Conclusion

In conclusion, the simplified expression 50−2\sqrt{50}-\sqrt{2} is equivalent to 424\sqrt{2}. This can be seen by breaking down the square root of 50, combining like terms, and evaluating the given choices.

Key Takeaways

  • Simplifying radical expressions involves combining like terms and removing any unnecessary radicals.
  • The expression 50−2\sqrt{50}-\sqrt{2} can be simplified to 424\sqrt{2}.
  • The correct answer is choice B, 424 \sqrt{2}.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

  • Mathway: A online math problem solver that can help you simplify radical expressions.
  • Khan Academy: A free online learning platform that offers video lessons and practice exercises on simplifying radical expressions.
  • Wolfram Alpha: A powerful online calculator that can help you simplify radical expressions and perform other mathematical operations.
    Simplifying Radical Expressions: A Q&A Guide ===========================================================

Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or other root. For example, 50\sqrt{50} is a radical expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break down the square root of the number inside the radical sign into its prime factors. Then, you can combine like terms and remove any unnecessary radicals.

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

A: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as 424^2. A non-perfect square is a number that cannot be expressed as the square of an integer.

Q: How do I simplify a radical expression with a non-perfect square?

A: To simplify a radical expression with a non-perfect square, you need to break down the square root of the number inside the radical sign into its prime factors. Then, you can combine like terms and remove any unnecessary radicals.

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. For example, 3/4 is a rational number. An irrational number is a number that cannot be expressed as the ratio of two integers. For example, 2\sqrt{2} is an irrational number.

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

A: To simplify a radical expression with an irrational number, you need to break down the square root of the number inside the radical sign into its prime factors. Then, you can combine like terms and remove any unnecessary radicals.

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

A: A radical is an expression that contains a square root or other root. For example, 50\sqrt{50} is a radical expression. An exponent is an expression that contains a power or exponent. For example, 232^3 is an exponent expression.

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

A: To simplify a radical expression with an exponent, you need to break down the square root of the number inside the radical sign into its prime factors. Then, you can combine like terms and remove any unnecessary radicals.

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

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

  • Not breaking down the square root of the number inside the radical sign into its prime factors.
  • Not combining like terms.
  • Not removing unnecessary radicals.
  • Not checking for rational or irrational numbers.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises. You can also use online resources such as math problem solvers or video lessons to help you practice.

Conclusion

In conclusion, simplifying radical expressions involves breaking down the square root of the number inside the radical sign into its prime factors, combining like terms, and removing any unnecessary radicals. By following these steps and avoiding common mistakes, you can simplify radical expressions with ease.

Key Takeaways

  • Simplifying radical expressions involves breaking down the square root of the number inside the radical sign into its prime factors.
  • Combining like terms and removing unnecessary radicals are essential steps in simplifying radical expressions.
  • Rational and irrational numbers can affect the simplification of radical expressions.
  • Exponents and radicals are different mathematical concepts.
  • Practicing simplifying radical expressions is essential to mastering this skill.

Additional Resources

For more information on simplifying radical expressions, check out the following resources:

  • Mathway: A online math problem solver that can help you simplify radical expressions.
  • Khan Academy: A free online learning platform that offers video lessons and practice exercises on simplifying radical expressions.
  • Wolfram Alpha: A powerful online calculator that can help you simplify radical expressions and perform other mathematical operations.