Simplify Nested Square Root: $\boldsymbol{\sqrt{9+4\sqrt{5}}}$

The question asks us to simplify the expression $\sqrt{9+4\sqrt{5}}$. This is a problem involving a nested radical (a square root within a square root).

Method for Simplifying Nested Radicals

To simplify $\sqrt{A + B\sqrt{C}}$ or similar forms, we often try to express the term inside the square root, $A + B\sqrt{C}$, as the square of a binomial, typically in the form $(x+y)^2 = x^2 + y^2 + 2xy$. Specifically, we aim for the form $(\sqrt{a} + \sqrt{b})^2$ or $(a + \sqrt{b})^2$.

Recall the identity: $(\sqrt{a} + \sqrt{b})^2 = (\sqrt{a})^2 + (\sqrt{b})^2 + 2\sqrt{a}\sqrt{b} = a + b + 2\sqrt{ab}$.

Our expression is $\sqrt{9+4\sqrt{5}}$. Let's rewrite the term $4\sqrt{5}$ to fit the form $2\sqrt{ab}$.

$$ 4\sqrt{5} = 2 \times 2\sqrt{5} $$

To bring the '2' inside the square root, we square it:

$$ 4\sqrt{5} = 2 \sqrt{2^2 \times 5} = 2 \sqrt{4 \times 5} = 2\sqrt{20} $$

So, the expression becomes:

$$ \sqrt{9 + 2\sqrt{20}} $$

Finding Suitable Numbers

Now, we compare this with the formula $a + b + 2\sqrt{ab}$. We need to find two numbers, $a$ and $b$, such that:

• Their sum ($a+b$) equals 9 (the number outside the radical).
• Their product ($a \times b$) equals 20 (the number inside the radical).

Let's list the pairs of factors for 20:

• 1 and 20 (Sum: $1+20=21$)
• 2 and 10 (Sum: $2+10=12$)
• 4 and 5 (Sum: $4+5=9$)

The pair 4 and 5 satisfies both conditions: $4+5=9$ and $4 \times 5=20$. So, we can set $a=5$ and $b=4$ (or vice versa).

Applying the Formula and Simplifying

Using $a=5$ and $b=4$, we can rewrite the expression inside the square root:

$$ 9 + 2\sqrt{20} = (5 + 4) + 2\sqrt{5 \times 4} = (\sqrt{5} + \sqrt{4})^2 $$

Now substitute this back into the original square root:

$$ \sqrt{9+4\sqrt{5}} = \sqrt{9 + 2\sqrt{20}} = \sqrt{(\sqrt{5} + \sqrt{4})^2} $$

Since the square root of a squared number is the absolute value of the number ($\sqrt{x^2} = |x|$), and $\sqrt{5} + \sqrt{4}$ is positive:

$$ \sqrt{(\sqrt{5} + \sqrt{4})^2} = \sqrt{5} + \sqrt{4} $$

We know that $\sqrt{4} = 2$. Therefore, the simplified expression is:

$$ \sqrt{5} + 2 $$

This is commonly written as $2 + \sqrt{5}$.

Verification

To check our answer, let's square $2 + \sqrt{5}$:

$$ (2 + \sqrt{5})^2 = 2^2 + (\sqrt{5})^2 + 2(2)(\sqrt{5}) $$

$$ = 4 + 5 + 4\sqrt{5} $$

$$ = 9 + 4\sqrt{5} $$

This matches the original expression inside the square root, confirming that $2 + \sqrt{5}$ is the correct simplification.

Conclusion

The simplified form of $\sqrt{9+4\sqrt{5}}$ is $2 + \sqrt{5}$.