Question

question 1
consider the expression below.
\\(\left(\sqrt{x^3 y z^2}\
ight)\left(x \sqrt{y^3}\
ight)\\)
which is equivalent to this expression?
answer
\\(x^{\frac{1}{2}} y^2 z\\) \\(x^{\frac{5}{2}} y^2 z\\)
\\(x^{\frac{3}{2}} y^{\frac{3}{4}} z\\) \\(x^{\frac{5}{2}} y^{\frac{5}{2}} z^{\frac{1}{2}}\\)

Explanation:

Step1: Convert radicals to exponents

Recall that $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. So, $\sqrt{x^3yz^2}=(x^3yz^2)^{\frac{1}{2}}$ and $\sqrt{y^3}=(y^3)^{\frac{1}{2}}$. The expression becomes:
$$(x^3yz^2)^{\frac{1}{2}}\cdot x\cdot (y^3)^{\frac{1}{2}}$$

Step2: Apply exponent power rule $(a^m)^n = a^{mn}$

For $(x^3yz^2)^{\frac{1}{2}}$, we get $x^{\frac{3}{2}}y^{\frac{1}{2}}z^{2\cdot\frac{1}{2}} = x^{\frac{3}{2}}y^{\frac{1}{2}}z$. For $(y^3)^{\frac{1}{2}}$, we get $y^{\frac{3}{2}}$. Now the expression is:
$$x^{\frac{3}{2}}y^{\frac{1}{2}}z\cdot x\cdot y^{\frac{3}{2}}$$

Step3: Apply exponent product rule $a^m \cdot a^n = a^{m + n}$

For $x$ terms: $x^{\frac{3}{2}} \cdot x^1 = x^{\frac{3}{2}+1}=x^{\frac{5}{2}}$. For $y$ terms: $y^{\frac{1}{2}} \cdot y^{\frac{3}{2}} = y^{\frac{1}{2}+\frac{3}{2}}=y^{2}$. The $z$ term remains $z$. So the simplified expression is:
$$x^{\frac{5}{2}}y^{2}z$$

Answer:

$x^{\frac{5}{2}}y^{2}z$ (corresponding to the option with this expression)