Rationalize the denominator.

StartFraction 5 Over StartRoot 5 EndRoot plus 2 EndFraction
Question content area bottom
Part 1
StartFraction 5 Over StartRoot 5 EndRoot plus 2 EndFraction
equals
  
enter your response here
​(Type an exact​ answer, using radicals as needed. Simplify your answer. Rationalize all​ denominators.)

1 answer

To rationalize the denominator of the expression

\[ \frac{5}{\sqrt{5} + 2}, \]

we can multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{5} - 2\). This technique eliminates the square root in the denominator.

Let's perform the calculation step-by-step:

  1. Multiply the numerator and denominator by the conjugate:

\[ \frac{5}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{5(\sqrt{5} - 2)}{(\sqrt{5} + 2)(\sqrt{5} - 2)}. \]

  1. Calculate the denominator using the difference of squares:

\[ (\sqrt{5} + 2)(\sqrt{5} - 2) = (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1. \]

  1. Rewrite the expression:

Now we have:

\[ \frac{5(\sqrt{5} - 2)}{1} = 5(\sqrt{5} - 2). \]

  1. Distributing the numerator:

Distributing gives:

\[ 5\sqrt{5} - 10. \]

Thus, the rationalized form of the original expression \(\frac{5}{\sqrt{5} + 2}\) is:

\[ 5\sqrt{5} - 10. \]

So the final answer is:

\[ \boxed{5\sqrt{5} - 10}. \]

Similar Questions
  1. Rationalize the denominator and simplify.StartFraction StartRoot 72 EndRoot Over StartRoot 7 EndRoot EndFraction Question
    1. answers icon 1 answer
    1. answers icon 1 answer
  2. Use the square root method to solve the equation.y squaredminus8equals0 Question content area bottom Part 1 A.4 4 B.plus or
    1. answers icon 1 answer
  3. Solve the equation by completing the square.zsquaredplus 10zplus 6equals0 Question content area bottom Part 1 A.5 plus or minus
    1. answers icon 1 answer
more similar questions