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 $10$. For very large $a$ and small $b$ (where $a\gg b$), $\sqrt{a + b}\approx\sqrt{a}$ because the contribution of $b$ to the sum $a + b$ is negligible. Here, $a = 985^{9}$ and $b = 10$, and $985^{9}$ is astronomically larger than $10$.

Step2: Approximate the Square Root

So, $\sqrt{985^{9}+10}\approx\sqrt{985^{9}}$. Using the property of exponents and square roots, $\sqrt{x^{n}}=x^{\frac{n}{2}}$ (for $x\geq0$ and $n$ an integer). Here, $x = 985$ and $n = 9$, so $\sqrt{985^{9}}=985^{\frac{9}{2}}=985^{4.5}$. We can also write this as $985^{4}\times\sqrt{985}$. But since we are looking for an approximation, and noting that $985$ is close to $1000$, but a better approximation is to recognize that for large $x$, $\sqrt{x^{9}+c}\approx x^{\frac{9}{2}}$ where $c$ is a constant (here $c = 10$). However, if we want a numerical approximation, we can note that $985^{9}$ is $(985^{4})^{2}\times985$, so $\sqrt{985^{9}}=985^{4}\times\sqrt{985}$. Calculating $985^{2}=970225$, $985^{4}=(985^{2})^{2}=970225^{2}=941336500625$, and $\sqrt{985}\approx31.38$. Then $941336500625\times31.38\approx941336500625\times30 + 941336500625\times1.38\approx2.824009501875\times10^{13}+1.2990443708625\times10^{13}\approx4.1230538727375\times10^{13}$. But a simpler approximation is to see that $\sqrt{985^{9}+10}\approx985^{4.5}\approx(1000 - 15)^{4.5}$. But the key idea is that the $+ 10$ has almost no effect on the square root of $985^{9}$. So the approximation is very close to $985^{4.5}$ or $985^{4}\sqrt{985}$.

Answer:

$\approx985^{4.5}$ (or approximately $4.12\times10^{13}$ when calculated numerically, but the main approximation is based on neglecting the $+10$ term due to the dominance of $985^{9}$)