Simplify The Following Expression:$\frac{5}{\sqrt{10}}$Options:A. $\sqrt{5}$ B. $\frac{\sqrt{2}}{2}$ C. $\sqrt{2}$ D. $- \frac{\sqrt{10}}{2}$

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Rationalizing the Denominator: Simplifying the Expression 510\frac{5}{\sqrt{10}}

In mathematics, rationalizing the denominator is a process used to eliminate any radicals from the denominator of a fraction. This is particularly useful when dealing with expressions that involve square roots or other types of radicals. In this article, we will explore how to simplify the expression 510\frac{5}{\sqrt{10}} by rationalizing the denominator.

The given expression is 510\frac{5}{\sqrt{10}}. Our goal is to simplify this expression by rationalizing the denominator. To do this, we need to get rid of the radical in the denominator. We can achieve this by multiplying both the numerator and the denominator by a suitable expression that will eliminate the radical.

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 10\sqrt{10} is 10\sqrt{10}. By multiplying both the numerator and the denominator by 10\sqrt{10}, we can eliminate the radical in the denominator.

510×1010=51010\frac{5}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{5\sqrt{10}}{10}

Now that we have rationalized the denominator, we can simplify the expression further. We can simplify the fraction 51010\frac{5\sqrt{10}}{10} by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

51010=102\frac{5\sqrt{10}}{10} = \frac{\sqrt{10}}{2}

In conclusion, we have successfully simplified the expression 510\frac{5}{\sqrt{10}} by rationalizing the denominator. By multiplying both the numerator and the denominator by the conjugate of the denominator, we were able to eliminate the radical in the denominator and simplify the expression further.

The correct answer is 102\boxed{\frac{\sqrt{10}}{2}}. This is the simplified form of the expression 510\frac{5}{\sqrt{10}} after rationalizing the denominator.

Let's compare our answer with the options provided:

  • Option A: 5\sqrt{5} - This is not the correct answer.
  • Option B: 22\frac{\sqrt{2}}{2} - This is not the correct answer.
  • Option C: 2\sqrt{2} - This is not the correct answer.
  • Option D: −102- \frac{\sqrt{10}}{2} - This is not the correct answer.

Our answer, 102\frac{\sqrt{10}}{2}, is the only option that matches the simplified form of the expression 510\frac{5}{\sqrt{10}} after rationalizing the denominator.

Rationalizing the denominator is an important concept in mathematics that can be used to simplify expressions involving radicals. By following the steps outlined in this article, you can simplify expressions like 510\frac{5}{\sqrt{10}} and arrive at the correct answer. Remember to always multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the radical in the denominator.
Frequently Asked Questions (FAQs) about Rationalizing the Denominator

A: Rationalizing the denominator is a process used to eliminate any radicals from the denominator of a fraction. This is particularly useful when dealing with expressions that involve square roots or other types of radicals.

A: We need to rationalize the denominator to simplify expressions and make them easier to work with. By eliminating the radical in the denominator, we can perform operations such as addition, subtraction, multiplication, and division more easily.

A: To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a radical is the same radical with the opposite sign.

A: The conjugate of a radical is the same radical with the opposite sign. For example, the conjugate of 10\sqrt{10} is −10-\sqrt{10}.

A: To find the conjugate of a radical, simply change the sign of the radical. For example, the conjugate of 10\sqrt{10} is −10-\sqrt{10}.

A: Yes, you can rationalize the denominator of a fraction with a negative sign. Simply multiply both the numerator and the denominator by the conjugate of the denominator, just like you would with a positive sign.

A: If the denominator is a binomial, you can rationalize it by multiplying both the numerator and the denominator by the conjugate of the binomial. For example, if the denominator is a+ba + b, you can multiply both the numerator and the denominator by a−ba - b.

A: Yes, you can rationalize the denominator of a fraction with a variable. Simply multiply both the numerator and the denominator by the conjugate of the denominator, just like you would with a numerical value.

A: If you have a fraction with a radical in the numerator and a radical in the denominator, you can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

A: Yes, you can rationalize the denominator of a fraction with a complex number. Simply multiply both the numerator and the denominator by the conjugate of the denominator, just like you would with a numerical value.

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Not multiplying both the numerator and the denominator by the conjugate of the denominator
  • Not changing the sign of the radical when finding the conjugate
  • Not simplifying the expression after rationalizing the denominator

A: To check if you have rationalized the denominator correctly, simply multiply both the numerator and the denominator by the conjugate of the denominator and see if the radical in the denominator is eliminated. If it is, then you have rationalized the denominator correctly.