Why the result is not $0$?

Question

Applying some properties, this should be: $(\sqrt[n]{a} * \sqrt[k]{a} ) - (a^{\frac{n+k}{nk}})= 0$

But according to symbolab, not.

I guess it's a symbolab error. But I would like to make sure.

Answer 1

It is not a symbolab error.

If we have two powers $x^m$ and $x^n$, then $x^m\cdot x^n = x^{m+n}$. This is known as the Power Product Rule because we are taking the product of two powers, but with a common base $x$.

If $m = \dfrac{1}{i}$ and $n = \dfrac{1}{j}$ then the same is still applied, such that $x^{1/i}\cdot x^{1/j} = x^{1/i\,+\, 1/j}$.

When we raise $x$ to the power of a fraction with numerator $1$ and denominator $h$ (or the reciprocal of $h$), then this is the same as $\sqrt [h]x$. Therefore, this power rule also applies to roots (or radicals).

It follows, then, that symbolab makes no mistake in writing that $$\Large \left(\sqrt [n] a\cdot \sqrt[k] a\right) - \left(a^{(n+k)/nk}\right) = a^{1/n\, + \, 1/k} - a^{(n+k)/nk}.$$

But it is technically incorrect as it should equal $0$ since, $$\frac{n+k}{nk} = \frac{n}{nk} + \frac{k}{nk} = \frac{\require{\cancel}\cancel n}{\cancel nk} + \frac{\cancel k}{n\cancel k} = \frac 1k + \frac 1n.$$