Question

(1÷15)^(1+2√5)×5^(2+2√5)÷3^(2−2√5)

Answer 1

$\displaystyle \frac{\displaystyle \left(\frac{1}{15}\right)^{1+2\sqrt 5}\times5^{2+2\sqrt 5}}{3^{2-2\sqrt 5}}=\frac{\displaystyle \left(\frac{1}{3\cdot5}\right)^{1+2\sqrt 5}\times5^{2+2\sqrt 5}}{3^{2-2\sqrt 5 }}=\\ \\ \frac{ \displaystyle \left(\frac{1}{3\cdot 5}\right)\left( \frac{1}{3\cdot 5} \right)^{2\sqrt 5}\cdot 5^2\cdot 5^{2\sqrt 5} } {\displaystyle \frac{3^2}{3^{2\sqrt 5}}}=\frac{5^2\cdot 5^{2\sqrt 5}\cdot 3^{2\sqrt 5}}{3\cdot 5\cdot 3^{2\sqrt 5}\cdot 5^{2\sqrt 5}\cdot 3^2}=\frac{5}{27}$

Answer 2

To simplify the expression (1÷15)^(1+2√5)×5^(2+2√5)÷3^(2−2√5), we can break it down into individual terms and simplify each term separately.

Let's start with the first term, (1÷15)^(1+2√5):

(1÷15)^(1+2√5) = (1/15)^(1+2√5)

To simplify this term, we can rewrite 1/15 as a power of 15:

(1/15)^(1+2√5) = (15^(-1))^(1+2√5) = 15^(-(1+2√5))

Moving on to the second term, 5^(2+2√5):

We can rewrite 5 as a power of 3:

5^(2+2√5) = (3+2)^(2+2√5) = 3^(2+2√5) × 2^(2+2√5)

And for the third term, 3^(2−2√5):

Similarly, we can rewrite 3 as a power of 5:

3^(2−2√5) = (5−2)^(2−2√5) = 5^(2−2√5) × 2^(2−2√5)

Now let's substitute these simplified forms back into the original expression:

(1÷15)^(1+2√5)×5^(2+2√5)÷3^(2−2√5) = 15^(-(1+2√5)) × 3^(2+2√5) × 2^(2+2√5) ÷ (5^(2−2√5) × 2^(2−2√5))

Since we have common bases (2 and 15) in the numerator and denominator, we can simplify further:

= (15/2)^(-(1+2√5)) × 3^(2+2√5) ÷ 5^(2−2√5)

At this point, we have simplified the expression as much as possible without knowing the specific values of √5 and without using numerical approximations.