Problem 1: Verify whether the statement $\sqrt{3^2 + 4^2} = 3 + 4$ is true or false.
Problem 3: Verify the inequality $\sqrt{8^2 + 15^2} \neq 8 + 15$.
Problem 5: Verify the identity $\sqrt{(9 + 7)^2} = 9 + 7$, given that $a + b > 0$.
Problem 7: Verify whether the statement $\sqrt{7^2 – 5^2} = 7 – 5$ is true or false.
Problem 9: Evaluate and verify the inequality $\sqrt{11^2 – 6^2} \neq 11 – 6$.
Problem 10: State whether the statement $\sqrt{(a + b)^2} = a + b$ (where $a + b > 0$) is true or false and provide a reason.
Problem 2: Find the value of the L.H.S and R.H.S and compare: $\sqrt{5^2 + 12^2}$ and $5 + 12$.
Problem 4: Evaluate both sides and state whether they are equal or not: $\sqrt{(6 + 2)^2}$ and $6 + 2$.
Problem 6: Find the L.H.S and R.H.S and compare: $\sqrt{(10 + 3)^2}$ and $10 + 3$.
Problem 8: Compare the L.H.S and R.H.S: $\sqrt{9^2 – 4^2}$ and $9 – 4$
These problems demonstrate that $\sqrt{a^2 + b^2}$ is not equal to $a + b$. You must perform the addition inside the square root before taking the root.
Problem 1: $\sqrt{3^2 + 4^2} = 3 + 4$
Explanation: On the left, you first square the numbers ($9 + 16 = 25$) and then take the square root ($\sqrt{25} = 5$). On the right, $3 + 4 = 7$. Since $5 \neq 7$, the statement is False.
Problem 2: $\sqrt{5^2 + 12^2}$ and $5 + 12$
Explanation: Following the order of operations, the left side becomes $\sqrt{25 + 144} = \sqrt{169}$, which is $13$. The right side is $17$. Comparison: $13 < 17$.
Problem 3: $\sqrt{8^2 + 15^2} \neq 8 + 15$
Explanation: The left side is $\sqrt{64 + 225} = \sqrt{289}$, which equals $17$. The right side is $23$. Since $17$ does not equal $23$, the inequality is Verified.
These problems show that the square root and the exponent $2$ cancel each other out when they apply to the entire group.
Problem 4: $\sqrt{(6 + 2)^2}$ and $6 + 2$
Explanation: Because the addition $(6+2)$ is grouped inside the parentheses and then squared, the square root “undoes” the square. Both sides equal $8$.
Problem 5: $\sqrt{(9 + 7)^2} = 9 + 7$
Explanation: Similar to problem 4, $\sqrt{16^2}$ is simply $16$, and $9+7$ is also $16$. The identity is Verified.
Problem 6: $\sqrt{(10 + 3)^2}$ and $10 + 3$
Explanation: The square root of $13^2$ is $13$. Since both sides result in $13$, they are Equal.
Similar to addition, you cannot simply “distribute” a square root over subtraction.
Problem 7: $\sqrt{7^2 – 5^2} = 7 – 5$
Explanation: The left side is $\sqrt{49 – 25} = \sqrt{24}$ (approx $4.9$). The right side is $2$. Since $4.9 \neq 2$, the statement is False.
Problem 8: $\sqrt{9^2 – 4^2}$ and $9 – 4$
Explanation: The left side is $\sqrt{81 – 16} = \sqrt{65}$ (approx $8.06$). The right side is $5$. Comparison: $8.06 > 5$.
Problem 9: $\sqrt{11^2 – 6^2} \neq 11 – 6$
Explanation: The left side is $\sqrt{121 – 36} = \sqrt{85}$ (approx $9.22$). The right side is $5$. The inequality is Verified.
Problem 10: $\sqrt{(a + b)^2} = a + b$ (where $a + b > 0$)
Explanation: This is True. When you square a value and then take its square root, you return to the original value (as long as the value is positive). The parentheses ensure the root applies to the sum as a single unit.