1. **State the problem:** Simplify the expression $$\sqrt{7 - t} + \sqrt{7 + t}$$ and determine when its value is a natural number (i.e., an integer greater than or equal to 1). 2. **Simplify the expression:** Let $$x = \sqrt{7 - t} + \sqrt{7 + t}$$. 3. **Square both sides to eliminate the square roots:** $$x^2 = (\sqrt{7 - t} + \sqrt{7 + t})^2 = (7 - t) + (7 + t) + 2\sqrt{(7 - t)(7 + t)}$$ 4. **Simplify the terms inside:** $$(7 - t) + (7 + t) = 14$$ $$(7 - t)(7 + t) = 49 - t^2$$ 5. **Rewrite:** $$x^2 = 14 + 2\sqrt{49 - t^2}$$ 6. **Isolate the square root term:** $$x^2 - 14 = 2\sqrt{49 - t^2}$$ 7. **Square again to solve for t:** $$(x^2 - 14)^2 = 4(49 - t^2)$$ 8. **Expand and simplify:** $$x^4 - 28x^2 + 196 = 196 - 4t^2$$ 9. **Cancel 196 on both sides:** $$x^4 - 28x^2 = -4t^2$$ 10. **Rearranged:** $$4t^2 = 28x^2 - x^4$$ 11. **Simplify:** $$t^2 = \frac{28x^2 - x^4}{4} = 7x^2 - \frac{x^4}{4}$$ 12. **Conditions for $t$ and $x$:** - $t^2 \geq 0$ so $7x^2 - \frac{x^4}{4} \geq 0$ - Since $x$ represents $\sqrt{7 - t} + \sqrt{7 + t}$, $x \geq 0$ - We want $x \in \mathbb{N}$ (natural numbers) 13. **Analyze the inequality:** $$7x^2 - \frac{x^4}{4} \geq 0 \implies 28x^2 - x^4 \geq 0 \implies x^2(28 - x^2) \geq 0$$ 14. **From this,** $$x^2 \leq 28$$ So natural $x$ must satisfy $x^2 \leq 28$, i.e., $x \leq 5$. 15. **Check integer values $x=1,2,3,4,5$: compute corresponding $t^2$ and check if $t$ is real:** - $x=1$: $t^2 = 7(1) - 1/4 = 7 - 0.25 = 6.75$ real - $x=2$: $t^2 = 7(4) - 16/4 = 28 - 4 = 24$ real - $x=3$: $t^2 = 7(9) - 81/4 = 63 - 20.25 = 42.75$ real - $x=4$: $t^2 = 7(16) - 256/4 = 112 - 64 = 48$ real - $x=5$: $t^2 = 7(25) - 625/4 = 175 - 156.25 = 18.75$ real All correspond to real $t$ values. 16. **Finally, check domain restrictions:** Since $$\sqrt{7 - t}$$ and $$\sqrt{7 + t}$$ are real, we need $7 - t \geq 0 \Rightarrow t \leq 7$ and $7 + t \geq 0 \Rightarrow t \geq -7$ 17. **Check $t$ values for each $x$ (taking $t = \pm \sqrt{t^2}$) to confirm domain is satisfied.** All $t$ values from step 15 satisfy $|t| \leq 7$. **Answer:** The expression $$\sqrt{7 - t} + \sqrt{7 + t}$$ is a natural number $x$ for $x \in \{1,2,3,4,5\}$ with corresponding $t = \pm \sqrt{7x^2 - \frac{x^4}{4}}$ within the domain $[-7,7]$.