Question

State with reason, whether the following is a surd and which is not?

`root(3)(8) xx root(3)(32)`

Answer 1

Surd

Reason:

Given expression: `root(3)(8) xx root(3)(32)`

Step-wise calculation:

1. Using the law of surds: `root(n)(a) xx root(n)(b) = root(n)(a xx b)`

So, `root(3)(8) xx root(3)(32) = root(3)(8 xx 32)`

2. Calculate the product inside the cube root: 8 × 32 = 256

3. Express 256 as powers of primes: 256 = 2⁸

4. Simplify the cube root:

`root(3)(2^8) = 2^(8//3)`

= `2^(2 + 2/3)`

= `2^2 xx 2^(2/3)`

= `4 xx root(3)(4)`

5. Since `root(3)(4)` is irrational not expressible as a rational number, the full expression simplifies to `4 xx root(3)(4)` which is irrational.

`root(3)(8) xx root(3)(32) = 4 xx root(3)(4)` is not a rational number.

Therefore, `root(3)(8) xx root(3)(32)` is a surd because it cannot be simplified to a rational number.