Question

the value of (1003)1/3 according to Binomial Theorem

Answer 1

Given, $\bold{\sqrt[3]{1003}}$
We know, according to Binomial theorem,
if Any number in the form of (1 + a)ⁿ , where a << 1
Then, (1 + a)ⁿ ≈ 1 + na

Now, (1003)⅓ = (1000 + 3)⅓
= (1000)⅓[ 1 + 3/1000]⅓
= 10[1 + 0.003]⅓
∵ 0.003 << 1
so, 10(1 + 0.003)⅓ = 10(1 + 0.003 × 1/3) = 10(1 + 0.001)
= 10 × 1.001
= 10.01

Hence, $\bold{\sqrt[3]{1003}}$ = 10.01

Answer 2

Hello Dear.

Given ⇒
Number = (1003)^1/3 
 = $\bold{\sqrt[3]{1003}}$

Now, According to the Binomial Theorem, If any number is in the form of (a + 1)ⁿ, where a is very less than 1, then number [say (a + 1)ⁿ] is equals to (an + 1)

We know, $\sqrt[3]{1003}$  can be written as (1000 + 3)⅓
 ∴ (1000 + 3)⅓ = (1000)⅓ + [1 + 3/1000]⅓
 (1000 + 3)⅓ = $\sqrt[3]{1000}$ + [1 + 0.003]⅓
(1000 + 3)⅓ = 10[ 1 + 0.003]⅓

Now, In this we can see that the 0.003 is less than 1.
∴ Applying the Binomial Theorem.
∴ 10(1 + 0.003)⅓  = 10(0.003 × 1/3 + 1)
   = 10( 1 + 0.001)
   = 10 × 1.001
    = 10.01


Hence, the value of the $\sqrt[3]{1003}$ by using the binomial theorem is 10.01.


Hope it helps.