given: $p = 15 \\left(\\frac{1}{6}\\right)^2 \\left(\\frac{5}{6}\\right)^4$ solve for $p$. round your answer…

Explanation:

Step1: Calculate the powers

First, calculate (\left(\frac{1}{6}\right)^2) and (\left(\frac{5}{6}\right)^4). (\left(\frac{1}{6}\right)^2=\frac{1^2}{6^2}=\frac{1}{36}\approx0.0278) (\left(\frac{5}{6}\right)^4=\frac{5^4}{6^4}=\frac{625}{1296}\approx0.4823)

Step2: Multiply the terms together

Now, multiply (15), (\left(\frac{1}{6}\right)^2), and (\left(\frac{5}{6}\right)^4) together. (P = 15\times\frac{1}{36}\times\frac{625}{1296}) First, (15\times\frac{1}{36}=\frac{15}{36}=\frac{5}{12}\approx0.4167) Then, (\frac{5}{12}\times\frac{625}{1296}=\frac{3125}{15552}\approx0.2009) (alternatively, multiply all at once: (15\times0.0278\times0.4823)) (15\times0.0278 = 0.417) (0.417\times0.4823\approx0.201) (more accurately, using exact fractions or more precise decimals)

Wait, let's do it more accurately:

(\left(\frac{1}{6}\right)^2=\frac{1}{36}), (\left(\frac{5}{6}\right)^4=\frac{625}{1296})

So (P = 15\times\frac{1}{36}\times\frac{625}{1296})

(15\times\frac{1}{36}=\frac{15}{36}=\frac{5}{12})

(\frac{5}{12}\times\frac{625}{1296}=\frac{5\times625}{12\times1296}=\frac{3125}{15552}\approx0.2009) (wait, no, wait: (15\times\frac{1}{36}\times\frac{625}{1296}=15\times\frac{625}{36\times1296}=15\times\frac{625}{46656}=\frac{9375}{46656}\approx0.2009))

Wait, actually, let's compute it as:

(P = 15\times\left(\frac{1}{6}\right)^2\times\left(\frac{5}{6}\right)^4=15\times\frac{1^2\times5^4}{6^{2 + 4}}=15\times\frac{625}{6^6})

(6^6 = 46656)

So (15\times625 = 9375)

Then (P=\frac{9375}{46656}\approx0.2009)

Wait, but let's check with decimal calculations:

(\frac{1}{6}\approx0.1667), so (\left(\frac{1}{6}\right)^2\approx0.1667^2 = 0.0278)

(\frac{5}{6}\approx0.8333), so (\left(\frac{5}{6}\right)^4\approx0.8333^4)

(0.8333^2 = 0.6944), then (0.6944^2\approx0.4823)

Then (15\times0.0278 = 0.417)

(0.417\times0.4823\approx0.417\times0.4823)

(0.4\times0.4823 = 0.19292)

(0.017\times0.4823 = 0.0081991)

Adding them: (0.19292+0.0081991 = 0.2011191\approx0.201) when rounded to three decimal places.

Wait, but let's use a calculator for more precision:

(\left(\frac{1}{6}\right)^2 = 1/36 ≈ 0.0277778)

(\left(\frac{5}{6}\right)^4 = 625/1296 ≈ 0.4822531)

Then (15\times0.0277778 = 0.416667)

Then (0.416667\times0.4822531 ≈ 0.416667\times0.4822531)

Calculate that:

(0.4\times0.4822531 = 0.19290124)

(0.0166667\times0.4822531 ≈ 0.00803755)

Sum: (0.19290124 + 0.00803755 ≈ 0.20093879\approx0.201) when rounded to three decimal places.

Answer:

(0.201)