Question

What is the value of $(1525{)}^{0.2}$ to $2$ decimal places?

• A. $4.33$
• B. $4.36$
• C. $4.38$
• D. $4.30$

Answer 1

The correct option is A $4.33$

we need to find$(1525{)}^{0.2}=(1525{)}^{1/5}$
Consider $f\left(x\right)=x^5-1525$ & $f^{\prime }\left(x\right)=5x^4$
Let initial assumption $x_0=4.0$
So by Newton Rapshon method
Iteration $1$: $x_1=x_0-\frac{f\left(x_0\right)}{f^{\prime }\left(x_0\right)}=4-\frac{f\left(4\right)}{f^{\prime }\left(4\right)}$
$=4.0-\left(\frac{4^5-1525}{{5.4}^4}\right)=4+\frac{501}{1280}=4.39$
Iteration $2$: $x_2=4.39-\frac{105.50}{1857.06}=4.33$
Iteration $3$: $x_3=4.33+\frac{2.91}{1757.60}=4.33$
Hence $5$th root of $1525$ is $4.33$