$\sqrt{1^{1.5}}=1$ why?

Question

ok, so I know that $\sqrt{{1}^{1.5}}=1$ but can anyone tell me why it equals $1$? what is it called? I want to understand and I'm a visual learner does it have a video?

Answer 1

In this problem $\sqrt{1^{1.5}}$ can be written as $1^\frac{1.5}{2}$. So, note that $1^n$ (where n= real numbers) is always $1$.

$1$ power any number is always $ 1 $.

Answer 2

Forgetting all the branching business of complex analysis, we can write it equivalently as $1^{3/4}$. Since $1^{3}=1$, we get $1^{1/4}$. So we want to solve the equation $1^{1/4}=x$. This is equivalent to $x^4=1$ and we know that 1 satisfies it.

I'm not sure what "it" is in "what is it called".

Answer 3

The statement comes down to understanding that 1 to any power is 1. If you want a visual explanation for this, you can start with the definition of exponent.
First, $$a^n = \overbrace{a \times a \times a \times \cdots \times a}^{n \ times}.$$

This is true for any whole number $n$. Thus $$1^3 = 1 \times 1 \times 1 = 1.$$

The square root of $1$ is $1$ from the fact that $1 \times 1 = 1$ in the same way that the square root of $4$ is $2$ from the fact that $2 \times 2 = 4.$

Next $a^{b/c}$ is defined to be $\sqrt[c]{a^b}$. Thus $$1^{3/2} = \sqrt[2]{1^3} = \sqrt[2]{1} = 1.$$

Finally $$\sqrt{1^{1.5}} = \sqrt{1^{3/2}}= \sqrt{1} = 1.$$