Question

The given equation is: $(\sqrt{2}+1)^3 = {}^3C_0 + {}^3C_1(\sqrt{2}) + {}^3C_2(\sqrt{2})^2 + {}^3C_3(\sqrt{2})^3$

• A. The equation is true and both sides evaluate to $7 + 5\sqrt{2}$.
• B. The equation is false.
• C. The equation is true and both sides evaluate to $1 + 3\sqrt{2}$.
• D. The equation is true and both sides evaluate to $7 + 7\sqrt{2}$.

Answer 1

Answer: (A)

The equation is true and both sides evaluate to $7 + 5\sqrt{2}$.

Answer 2

The given equation presents a binomial expansion. The left-hand side (LHS) is $(\sqrt{2}+1)^3$. The right-hand side (RHS) is the binomial expansion of $(1+\sqrt{2})^3$.

Evaluating the LHS: Using the formula $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$, with $a=\sqrt{2}$ and $b=1$: $(\sqrt{2}+1)^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(1) + 3(\sqrt{2})(1)^2 + (1)^3$ $= 2\sqrt{2} + 3(2) + 3\sqrt{2} + 1$ $= 2\sqrt{2} + 6 + 3\sqrt{2} + 1$ $= 7 + 5\sqrt{2}$

Evaluating the RHS: The RHS is ${}^3C_0 + {}^3C_1(\sqrt{2}) + {}^3C_2(\sqrt{2})^2 + {}^3C_3(\sqrt{2})^3$. This is the binomial expansion of $(1+\sqrt{2})^3$. Calculating the binomial coefficients: ${}^3C_0 = 1$ ${}^3C_1 = 3$ ${}^3C_2 = 3$ ${}^3C_3 = 1$ Calculating the powers of $\sqrt{2}$: $(\sqrt{2})^1 = \sqrt{2}$ $(\sqrt{2})^2 = 2$ $(\sqrt{2})^3 = 2\sqrt{2}$

Substituting these values into the RHS: $RHS = 1(1) + 3(\sqrt{2}) + 3(2) + 1(2\sqrt{2})$ $RHS = 1 + 3\sqrt{2} + 6 + 2\sqrt{2}$ $RHS = 7 + 5\sqrt{2}$

Since LHS = RHS = $7 + 5\sqrt{2}$, the given equation is true.