Question

(3 - 9square root 7)(9 square root 13 - square root 2)

Answer 1

To simplify the expression \((3 - 9\sqrt{7})(9\sqrt{13} - \sqrt{2})\), we'll apply the distributive property (also known as the FOIL method in the case of two binomials).

Let's denote:
- \(a = 3\)
- \(b = -9\sqrt{7}\)
- \(c = 9\sqrt{13}\)
- \(d = -\sqrt{2}\)

Now, applying the formula \((a + b)(c + d) = ac + ad + bc + bd\):

1. **Compute \(ac\)**:
\[
ac = 3 \cdot 9\sqrt{13} = 27\sqrt{13}
\]

2. **Compute \(ad\)**:
\[
ad = 3 \cdot (-\sqrt{2}) = -3\sqrt{2}
\]

3. **Compute \(bc\)**:
\[
bc = -9\sqrt{7} \cdot 9\sqrt{13} = -81\sqrt{91}
\]
(since \(\sqrt{7} \cdot \sqrt{13} = \sqrt{91}\))

4. **Compute \(bd\)**:
\[
bd = -9\sqrt{7} \cdot (-\sqrt{2}) = 9\sqrt{14}
\]
(since \(\sqrt{7} \cdot \sqrt{2} = \sqrt{14}\))

Now, combining all the results:
\[
(3 - 9\sqrt{7})(9\sqrt{13} - \sqrt{2}) = 27\sqrt{13} - 3\sqrt{2} - 81\sqrt{91} + 9\sqrt{14}
\]

We can rearrange it to have a clearer structure:
\[
27\sqrt{13} + 9\sqrt{14} - 3\sqrt{2} - 81\sqrt{91}
\]

Thus, the final simplified expression is:
\[
27\sqrt{13} + 9\sqrt{14} - 3\sqrt{2} - 81\sqrt{91}
\]