Question

which expression is equivalent to \\(\sqrt{400h^8j^4k^8}\\)?
\\(\circ\\) \\(20h^4j^2|k^2|\\)
\\(\circ\\) \\(20h^6j^2k^4\\)
\\(\circ\\) \\(200h^4j^2|k^3|\\)
\\(\circ\\) \\(200h^6j^2k^4\\)

Explanation:

Step1: Simplify the square root of the constant term

The constant term is 400. We know that $\sqrt{400} = 20$ because $20\times20 = 400$.

Step2: Simplify the square root of the variable $h^8$

Using the property of square roots $\sqrt{x^n}=x^{\frac{n}{2}}$ (for $x\geq0$), for $h^8$, we have $\sqrt{h^8}=h^{\frac{8}{2}} = h^4$.

Step3: Simplify the square root of the variable $j^4$

For $j^4$, using the same property, $\sqrt{j^4}=j^{\frac{4}{2}}=j^2$.

Step4: Simplify the square root of the variable $k^6$

For $k^6$, $\sqrt{k^6}=k^{\frac{6}{2}}=k^3$. But since we are dealing with real numbers and the square root of a square is the absolute value, we write $|k^3|$. However, if we assume $k$ is a non - negative real number (which is a common assumption in many algebraic contexts unless stated otherwise), we can write $k^3$. But looking at the options, the first option has $|k^3|$ and the second has $k^4$ (wait, no, let's re - check the exponents). Wait, the original expression is $\sqrt{400h^{8}j^{4}k^{6}}$. Let's re - calculate the exponent of $k$: $\frac{6}{2}=3$, so $\sqrt{k^6}=|k^3|$. And for $h^8$: $\frac{8}{2} = 4$, so $h^4$; for $j^4$: $\frac{4}{2}=2$, so $j^2$; and $\sqrt{400}=20$. So putting it all together, we get $20h^{4}j^{2}|k^{3}|$.

Answer:

$20h^{4}j^{2}|k^{3}|$ (the first option: $20h^{4}j^{2}|k^{3}|$)