This worksheet helps students master rationalizing the denominator through a fun matching and riddle-solving activity.

Let’s solve this “Rationalizing the Denominator” worksheet step by step.

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Goal:

Simplify each expression by rationalizing the denominator (removing radicals from the denominator), then match the simplified form to the letter on the right. Finally, circle letters not crossed out and arrange them alphabetically to find the code.

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## Step 1: Rationalize Each Expression

We’ll go one by one:

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1. \( \frac{2}{\sqrt{3}} \)

Multiply numerator and denominator by \( \sqrt{3} \):

\[
\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\]

✔ Match: E (since E is at \( \frac{2\sqrt{3}}{3} \))

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2. \( \frac{4}{\sqrt{2}} \)

Multiply by \( \frac{\sqrt{2}}{\sqrt{2}} \):

\[
\frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}
\]

✔ Match: K (since K is at \( 2\sqrt{2} \))

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3. \( \frac{1}{\sqrt{3}} \)

Multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \):

\[
\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]

✔ Match: B (B is at \( \frac{\sqrt{3}}{3} \))

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4. \( \frac{5\sqrt{3}}{\sqrt{2}} \)

Multiply numerator and denominator by \( \sqrt{2} \):

\[
\frac{5\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{6}}{2}
\]

✔ Match: A (A is at \( \frac{5\sqrt{6}}{2} \))

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5. \( \frac{6\sqrt{2}}{\sqrt{6}} \)

Simplify first: \( \frac{6\sqrt{2}}{\sqrt{6}} = 6 \cdot \sqrt{\frac{2}{6}} = 6 \cdot \sqrt{\frac{1}{3}} = \frac{6}{\sqrt{3}} \)

Then rationalize:

\[
\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}
\]

✔ Match: F (F is at \( 2\sqrt{3} \))

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6. \( \frac{1}{\sqrt{2}} \)

Multiply by \( \frac{\sqrt{2}}{\sqrt{2}} \):

\[
\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

✔ Match: J (J is at \( \frac{\sqrt{2}}{2} \))

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7. \( \frac{9}{\sqrt{3}} \)

Multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \):

\[
\frac{9}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}
\]

✔ Match: T (T is at \( 3\sqrt{3} \))

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8. \( \frac{3\sqrt{27}}{\sqrt{3}} \)

First simplify \( \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} \), so:

\[
\frac{3 \cdot 3\sqrt{3}}{\sqrt{3}} = \frac{9\sqrt{3}}{\sqrt{3}} = 9
\]

✔ Match: W (W is at 9)

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9. \( \frac{20}{\sqrt{8}} \)

First simplify \( \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \), so:

\[
\frac{20}{2\sqrt{2}} = \frac{10}{\sqrt{2}}
\]

Now rationalize:

\[
\frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}
\]

✔ Match: M (M is at \( 5\sqrt{2} \))

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## Step 2: Cross Out the Matched Letters

From above, we matched:

- E → \( \frac{2\sqrt{3}}{3} \)
- K → \( 2\sqrt{2} \)
- B → \( \frac{\sqrt{3}}{3} \)
- A → \( \frac{5\sqrt{6}}{2} \)
- F → \( 2\sqrt{3} \)
- J → \( \frac{\sqrt{2}}{2} \)
- T → \( 3\sqrt{3} \)
- W → 9
- M → \( 5\sqrt{2} \)

So we cross out: A, B, E, F, J, K, M, T, W

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## Step 3: Find Letters NOT Crossed Out

Look at all letters on the grid:
A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z

Crossed out: A, B, E, F, J, K, M, T, W

Left:
C, D, G, H, I, L, N, O, P, Q, R, S, U, V, X, Y, Z

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## Step 4: Arrange Remaining Letters in Alphabetical Order

Alphabetical order:

C, D, G, H, I, L, N, O, P, Q, R, S, U, V, X, Y, Z

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## ✔ Final Answer: The Code is:

> CDGHILOPQRSUVXYZ

*(Note: Some worksheets may expect only a subset — but based on instructions, we include ALL uncrossed letters in alphabetical order.)*

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## 💡 Extra Tip:
This puzzle is designed to reinforce rationalizing denominators and matching simplified forms. The final “code” is just the leftover letters — often used for fun or to check work.

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✔ Answer: CDGHILOPQRSUVXYZ