Step 1: Find the value of $a + \frac{1}{a}$. Given $a = \frac{3+\sqrt{5}}{2}$. First, find $\frac{1}{a}$: $$ \frac{1}{a} = \frac{2}{3+\sqrt{5}} $$ Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, $3-\sqrt{5}$: $$ \frac{1}{a} = \frac{2}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}} = \frac{2(3-\sqrt{5})}{3^2 - (\sqrt{5})^2} = \frac{2(3-\sqrt{5})}{9-5} = \frac{2(3-\sqrt{5})}{4} = \frac{3-\sqrt{5}}{2} $$ Now, add $a$ and $\frac{1}{a}$: $$ a + \frac{1}{a} = \frac{3+\sqrt{5}}{2} + \frac{3-\sqrt{5}}{2} = \frac{3+\sqrt{5}+3-\sqrt{5}}{2} = \frac{6}{2} = 3 $$ Step 2: Use the identity $a^2 + \frac{1}{a^2} = \left(a + \frac{1}{a}\right)^2 - 2$. Substitute the value of $a + \frac{1}{a}$ found in Step 1: $$ a^2 + \frac{1}{a^2} = (3)^2 - 2 = 9 - 2 = 7 $$ The value of $a^2 + \frac{1}{a^2}$ is $\boxed{\text{7}}$. 63. Simplify: $(256)^{(-4^{-2})}$ Step 1: Evaluate the exponent $-4^{-2}$. $$ -4^{-2} = -\frac{1}{4^2} = -\frac{1}{16} $$ Step 2: Substitute the exponent back into the expression. $$ (256)^{(-1/16)} $$ Step 3: Express $256$ as a power of a base that simplifies with the exponent. We know that $256 = 2^8$. $$ (2^8)^{-1/16} = 2^{8 \times (-\frac{1}{16})} = 2^{-\frac{8}{16}} = 2^{-\frac{1}{2}} $$ Step 4: Simplify the expression. $$ 2^{-\frac{1}{2}} = \frac{1}{2^{1/2}} = \frac{1}{\sqrt{2}} $$ Rationalize the denominator: $$ \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$ The simplified expression is $\boxed{\frac{\sqrt{2}}{2}}$. 64. Find the value of $\frac{4}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{-\frac{3}{4}}} + \frac{2}{(243)^{-\frac{1}{5}}}$ Step 1: Rewrite the terms using the property $\frac{1}{x^{-n}} = x^n$. The expression becomes: $$ 4 \cdot (216)^{\frac{2}{3}} + 1 \cdot (256)^{\frac{3}{4}} + 2 \cdot (243)^{\frac{1}{5}} $$ Step 2: Evaluate each term. For the first term, $4 \cdot (216)^{\frac{2}{3}}$: Recognize that $216 = 6^3$. $$ 4 \cdot (6^3)^{\frac{2}{3}} = 4 \cdot 6^{3 \times \frac{2}{3}} = 4 \cdot 6^2 = 4 \cdot 36 = 144 $$ For the second term, $1 \cdot (256)^{\frac{3}{4}}$: Recognize that $256 = 4^4$. $$ 1 \cdot (4^4)^{\frac{3}{4}} = 1 \cdot 4^{4 \times \frac{3}{4}} = 1 \cdot 4^3 = 1 \cdot 64 = 64 $$ For the third term, $2 \cdot (243)^{\frac{1}{5}}$: Recognize that $243 = 3^5$. $$ 2 \cdot (3^5)^{\frac{1}{5}} = 2 \cdot 3^{5 \times \frac{1}{5}} = 2 \cdot 3^1 = 2 \cdot 3 = 6 $$ Step 3: Add the values of the terms. $$ 144 + 64 + 6 = 208 + 6 = 214 $$ The value of the expression is $\boxed{\text{214}}$. Send me the next one 📸