1. Given points and vectors in a coordinate system: $O(0,0)$, $I(1,0)$, $J(0,1)$ with $OI=OJ=1$ cm. 2. Points provided: $A(-3,0)$, $M(3,2)$, and $N(3,-2)$. 3. Determine if segment $MN$ is parallel to $OJ$: - Vector $MN = N - M = (3 - 3, -2 - 2) = (0, -4)$. - Vector $OJ = (0, 1)$. 4. Since $MN = (0, -4)$ is a scalar multiple of $OJ = (0, 1)$, $MN$ is parallel to $OJ$. 5. Review other points: - $A(\sqrt{2},201)$ and $B(-\sqrt{2}, | -201|) = (-\sqrt{2},201)$. - Point $A=5x7y$ likely means coordinates $(5,7)$ and $M=1a2a3a$ seems symbolic; assume indexing or variables. - Sets $E = \{ -\sqrt{3}, \frac{22}{7}, \frac{\sqrt{12}}{27}, \frac{\pi}{3}, \sqrt{0.90}, 0.12131415 \}$ involves numbers from irrational, rational and decimal types. 6. Find intersections of $E$ with sets $I, Z, Q$: - $E \cap I$: intersection with integers $I$; assess elements: - $-\sqrt{3} \approx -1.732$ (not integer) - $\frac{22}{7} \approx 3.142857$ (not integer) - $\frac{\sqrt{12}}{27} = \frac{2\sqrt{3}}{27} \approx 0.128$ (not integer) - $\frac{\pi}{3} \approx 1.047$ (not integer) - $\sqrt{0.90} \approx 0.9487$ (not integer) - $0.12131415$ decimal (not integer) => $E \cap I = \emptyset$. - $E \cap Z$: intersection with integers $Z$ (integers including negatives), same as above, no integer values. - $E \cap Q$: intersection with rationals $Q$; fractions like $\frac{22}{7}$ are rational, decimals like $0.12131415$ are rational if finite or repeating. => $E \cap Q = \{ \frac{22}{7}, 0.12131415 \}$. 7. Calculate $7 \times 27^8 - 23 \times 3^{22}$: - $27 = 3^3$, so $27^8 = (3^3)^8 = 3^{24}$. - $7 \times 27^8 = 7 \times 3^{24}$. - $3^{22}$ is given. - Expression becomes $7 \times 3^{24} - 23 \times 3^{22} = 3^{22}(7 \times 3^2 - 23) = 3^{22}(7 \times 9 - 23) = 3^{22}(63 - 23) = 3^{22} \times 40$. Final answer: $7 \times 27^8 - 23 \times 3^{22} = 40 \times 3^{22}$. Summary: - Vector MN is parallel to OJ. - $E \cap I = \emptyset$. - $E \cap Z = \emptyset$. - $E \cap Q = \{ \frac{22}{7}, 0.12131415 \}$. - Simplification of given exponential expression is $40 \times 3^{22}$.