1. **Problem Statement:** We need to verify the correctness of several mathematical statements and provide reasons. 2. **Statement 1:** "The numbers 2024 and 1446 are coprime." - Two numbers are coprime if their greatest common divisor (gcd) is 1. - Calculate $\gcd(2024,1446)$. 3. **Calculating gcd(2024,1446):** - Use the Euclidean algorithm: $$2024 = 1446 \times 1 + 578$$ $$1446 = 578 \times 2 + 290$$ $$578 = 290 \times 1 + 288$$ $$290 = 288 \times 1 + 2$$ $$288 = 2 \times 144 + 0$$ - The gcd is the last nonzero remainder, which is $2$. - Since $\gcd(2024,1446) = 2 \neq 1$, the numbers are **not coprime**. 4. **Statement 2:** "495 is the fractional number 4.98 of 99." - Check if $495 = 4.98 \times 99$. - Calculate $4.98 \times 99$: $$4.98 \times 99 = 4.98 \times (100 - 1) = 4.98 \times 100 - 4.98 = 498 - 4.98 = 493.02$$ - Since $495 \neq 493.02$, the statement is **false**. 5. **Statement 3:** "Fractional form: $-1 - \sqrt{2} \neq 1 - \sqrt{2}$." - Compare the two expressions: $$-1 - \sqrt{2} \quad \text{and} \quad 1 - \sqrt{2}$$ - They differ by 2, so they are **not equal**. - The statement is **true**. 6. **Statement 4:** For every real number $x \neq \sqrt{2}$, the expression: $$\left(\sqrt{2} - (\sqrt{2} - 1)^{1441}\right) \times (4\sqrt{3} - 7)^{1447} \times (4\sqrt{3} + 7)^{1447} = -\frac{1}{4}$$ - First, note that $(4\sqrt{3} - 7)$ and $(4\sqrt{3} + 7)$ are conjugates. - Their product is: $$ (4\sqrt{3} - 7)(4\sqrt{3} + 7) = (4\sqrt{3})^2 - 7^2 = 16 \times 3 - 49 = 48 - 49 = -1 $$ - Therefore: $$ (4\sqrt{3} - 7)^{1447} \times (4\sqrt{3} + 7)^{1447} = (-1)^{1447} = -1 $$ - So the expression simplifies to: $$ \left(\sqrt{2} - (\sqrt{2} - 1)^{1441}\right) \times (-1) = -\left(\sqrt{2} - (\sqrt{2} - 1)^{1441}\right) $$ - We want to check if this equals $-\frac{1}{4}$, so: $$ -\left(\sqrt{2} - (\sqrt{2} - 1)^{1441}\right) = -\frac{1}{4} $$ $$ \Rightarrow \sqrt{2} - (\sqrt{2} - 1)^{1441} = \frac{1}{4} $$ - This is a specific numeric equality that can be verified numerically or by further algebraic manipulation. - Since $(\sqrt{2} - 1) \approx 0.4142 < 1$, raising it to a large power $1441$ makes it very small. - So: $$ (\sqrt{2} - 1)^{1441} \approx 0 $$ - Hence: $$ \sqrt{2} - (\sqrt{2} - 1)^{1441} \approx \sqrt{2} \approx 1.4142 \neq \frac{1}{4} $$ - Therefore, the equality is **false**. **Final answers:** - Statement 1: False (2024 and 1446 are not coprime). - Statement 2: False (495 is not 4.98 times 99). - Statement 3: True ($-1 - \sqrt{2} \neq 1 - \sqrt{2}$). - Statement 4: False (expression does not equal $-\frac{1}{4}$).