Question

√(985⁹ + 10)

Explanation:

Step1: Analyze the Dominant Term

In the expression $\sqrt{985^9 + 10}$, the term $985^9$ is extremely large compared to $10$. So, $985^9 + 10 \approx 985^9$ for approximation purposes.

Step2: Simplify the Square Root

Using the approximation from Step 1, we have $\sqrt{985^9 + 10} \approx \sqrt{985^9}$. We can rewrite $985^9$ as $(985^{\frac{9}{2}})^2$, so $\sqrt{985^9}=985^{\frac{9}{2}}$. We can also write $985^{\frac{9}{2}}$ as $985^{4.5}=985^4\times\sqrt{985}$.
First, calculate $985^2 = (1000 - 15)^2=1000^2 - 2\times1000\times15 + 15^2 = 1000000 - 30000 + 225 = 970225$.
Then, $985^4=(985^2)^2 = 970225^2$. Let's calculate that: $970225\times970225$. But maybe we can also note that $985^{\frac{9}{2}} = 985^{4}\times\sqrt{985}\approx985^{4}\times31.38$. However, if we want a more precise or if we consider that for very large $n$, $\sqrt{n^k + m}\approx n^{\frac{k}{2}}$ when $m\ll n^k$. So, $\sqrt{985^9 + 10}\approx985^{\frac{9}{2}} = 985^{4}\times\sqrt{985}$. But if we just want to express it in terms of the original number's power, or if we consider that $985^9=(985^4)^2\times985$, so $\sqrt{985^9}=985^4\times\sqrt{985}$. Alternatively, we can write $985^9 = 985^{8 + 1}= (985^4)^2\times985$, so square root of that is $985^4\times\sqrt{985}$.
But if we are looking for an approximate value, since $985^9$ is much larger than $10$, the square root of $985^9 + 10$ is very close to $985^{\frac{9}{2}}$. Let's compute $985^{\frac{9}{2}}$:
$985^{\frac{1}{2}}\approx31.3847$
$985^1 = 985$
$985^2 = 970225$
$985^3 = 985^2\times985 = 970225\times985 = 970225\times(1000 - 15)=970225000 - 14553375 = 955671625$
$985^4 = 985^3\times985 = 955671625\times985 = 955671625\times(1000 - 15)=955671625000 - 14335074375 = 941336550625$
Then $985^4\times\sqrt{985}\approx941336550625\times31.3847\approx941336550625\times30 + 941336550625\times1.3847\approx28240096518750+1303000000000\approx29543096518750$ (this is a rough approximation). But the key idea is that the $+10$ is negligible, so the expression is approximately equal to $985^{\frac{9}{2}}$ or $985^4\sqrt{985}$.

Answer:

$\approx 985^{\frac{9}{2}}$ (or approximately $2.95\times 10^{25}$ when calculated numerically, but the exact simplified form considering the dominant term is $\approx 985^{\frac{9}{2}}$)