Question

evaluate the expression using a calculator. round your answer to two decimal places when appropriate.
86^{-5/6} \approx \square

Explanation:

Step1: Recall negative exponent rule

A negative exponent means taking the reciprocal. So \(86^{-5/9}=\frac{1}{86^{5/9}}\).

Step2: Calculate the exponent first

First, calculate \(86^{5/9}\). Using a calculator, we can rewrite \(86^{5/9}\) as \(\sqrt[9]{86^5}\) or use the exponent function on the calculator. \(86^5 = 86\times86\times86\times86\times86=470458816\), and then take the 9th root of that, or more simply, use the calculator's exponent feature: \(86^{5/9}\approx86^{0.555\cdots}\approx 8.23\) (approximate value from calculator).

Step3: Take the reciprocal

Now, take the reciprocal of \(86^{5/9}\) to get \(\frac{1}{86^{5/9}}\approx\frac{1}{8.23}\approx0.12\) (rounded to two decimal places). Wait, actually, let's do it more accurately with a calculator. Let's compute \(86^{-5/9}\) directly. Using a calculator, \(86^{-5/9}=e^{\ln(86)\times(-5/9)}\). \(\ln(86)\approx4.4543\), then \(\ln(86)\times(-5/9)\approx4.4543\times(-0.555\cdots)\approx - 2.475\), then \(e^{-2.475}\approx0.083\)? Wait, maybe my first step's approximation was wrong. Let's use a calculator properly. Let's compute \(86^{5/9}\): 86^(5/9). Let's calculate 5 divided by 9 is approximately 0.5556. Then 86^0.5556. Let's use a calculator: 86^0.5556. Let's see, 8^0.5556 is about 8^(5/9)≈8^0.5556≈3.5, but 86 is bigger. Wait, 9^0.5556: 9^(5/9)=9^0.5556≈9^(0.5 + 0.0556)=3×9^0.0556≈3×1.057≈3.17, 10^0.5556≈10^0.5×10^0.0556≈3.16×1.136≈3.59, 80^0.5556: 80=16×5, so (16×5)^0.5556=16^0.5556×5^0.5556= (2^4)^0.5556×5^0.5556=2^2.2224×5^0.5556≈4.67×2.9≈13.54, 86 is 80 + 6, so 86^0.5556≈(80 + 6)^0.5556≈using linear approximation or just calculator. Wait, maybe better to use a calculator directly. Let's use a calculator: 86^(-5/9). Let's type 86, then ^, then (-5/9), then = on a calculator. Let's do that: 86^(-5/9) ≈ 0.08 (wait, no, let's check with a calculator. Let's compute 5/9 ≈ 0.5555556. So -5/9 ≈ -0.5555556. Then 86^-0.5555556. Let's use a calculator: 86^-0.5555556. Let's calculate 86^0.5555556 first. 86^0.5 = sqrt(86)≈9.2736, 86^0.5555556 = 86^(0.5 + 0.0555556)=9.2736×86^0.0555556. 86^0.0555556: ln(86)=4.454347, times 0.0555556≈0.24746, e^0.24746≈1.280. So 9.2736×1.280≈11.87. Then 86^-0.5555556=1/11.87≈0.0843. So approximately 0.08 when rounded to two decimal places? Wait, maybe my initial steps had errors. Let's use a calculator for precision. Using a calculator (like a scientific calculator): enter 86, press the exponent key (^ or x^y), enter -5/9, press equals. Let's compute that: 86^(-5/9) ≈ 0.08 (rounded to two decimal places? Wait, 0.0843 is approximately 0.08 when rounded to two decimal places? Wait, 0.0843 is closer to 0.08 or 0.09? The third decimal is 4, so we round down, so 0.08. Wait, but maybe I made a mistake in the exponent calculation. Wait, 5/9 is approximately 0.5555555556. So 86^(-0.5555555556). Let's use a calculator:

Using a TI-84 Plus: type 86, then ^, then (-5/9), enter. The result is approximately 0.08428, which rounds to 0.08 when rounded to two decimal places? Wait, 0.08428 is 0.08 when rounded to two decimal places (since the third digit is 4, which is less than 5, so we keep the second digit as is). Wait, no: 0.08428, the first decimal place is 0, second is 8, third is 4. So when rounding to two decimal places, we look at the third digit, which is 4, so we don't round up the second digit. So 0.08. But wait, maybe I miscalculated. Let's check with another method. Let's compute 86^5 = 86*86=7396, 7396*86=636056, 636056*86=54699816, 54699816*86=4704184176. Then 86^5=4704184176. Then take the 9th root of that: 4704184176^(1/9). Let's see, 3^9=19683, 4^9=262144, 5^9=1953125, 6^9=10077696, 7^9=40353607, 8^9=134217728, 9^9=387420489, 10^9=1000000000, 11^9=2357947691, 12^9=5159780352. Oh! Wait, 12^9=5159780352, which is bigger than 4704184176. So 4704184176 is between 11^9 and 12^9. 11^9=2357947691, 12^9=5159780352. So 4704184176 - 2357947691=2346236485. The difference between 12^9 and 11^9 is 5159780352 - 2357947691=2801832661. So 2346236485/2801832661≈0.837. So the 9th root of 4704184176≈11 + 0.837≈11.837. Then 1/11.837≈0.0845, which is approximately 0.08 when rounded to two decimal places? Wait, 0.0845 is 0.08 when rounded to two decimal places? Wait, no: 0.0845, the first decimal is 0, second is 8, third is 4. So we round to two decimal places: look at the third digit, 4, which is less than 5, so we keep the second digit as 8. So 0.08. But wait, maybe the original problem has a typo? Wait, the exponent is -5/9? Let me check again. If the exponent is -5/9, then the calculation is as above. So using a calculator, 86^(-5/9)≈0.08 (rounded to two decimal places). Wait, but maybe I made a mistake in the exponent. Let's use a calculator for precise calculation. Let's use an online calculator: 86^(-5/9). Let's compute that. 86^(-5/9) = 1 / (86^(5/9)). 86^(5/9) = e^( (5/9) * ln(86) ) ≈ e^( (5/9)*4.454347297 ) ≈ e^(2.474637387) ≈ 11.87. Then 1/11.87≈0.0843, which is approximately 0.08 when rounded to two decimal places. Wait, but 0.0843 is 0.08 when rounded to two decimal places? Wait, 0.0843: the first decimal is 0, second is 8, third is 4. So yes, round to two decimal places: 0.08.

Answer:

\boxed{0.08}