Question

Blank radical 12 + 2 radical 27 = 14 radical blank

Answer 1

To solve the equation \( \sqrt{12} + 2\sqrt{27} = 14\sqrt{x} \), we start by simplifying each radical term.

1. **Simplifying \( \sqrt{12} \)**:
\[
\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}
\]

2. **Simplifying \( 2\sqrt{27} \)**:
\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\]
Therefore,
\[
2\sqrt{27} = 2 \cdot 3\sqrt{3} = 6\sqrt{3}
\]

3. **Combining the simplified terms**:
\[
\sqrt{12} + 2\sqrt{27} = 2\sqrt{3} + 6\sqrt{3} = (2 + 6)\sqrt{3} = 8\sqrt{3}
\]

Now, we have the left side simplified:
\[
8\sqrt{3} = 14\sqrt{x}
\]

4. **Setting the equations equal**:
\[
8\sqrt{3} = 14\sqrt{x}
\]

5. **Dividing both sides by 14**:
\[
\sqrt{x} = \frac{8}{14}\sqrt{3} = \frac{4}{7}\sqrt{3}
\]

6. **Squaring both sides to solve for \( x \)**:
\[
x = \left( \frac{4}{7} \right)^2 \cdot 3 = \frac{16}{49} \cdot 3 = \frac{48}{49}
\]

Thus, the blank in the expression \( 14\sqrt{\text{blank}} \) should be filled with:
\[
\boxed{\frac{48}{49}}
\]