Question

Question!
Find the value of :
$\displaystyle \rm \: \sqrt{ \frac{ {x}^{ {y}^{2} } {y}^{ {x}^{2} } {z}^{(x + y)} }{ {(x + y)}^{ {(x + y)}^{2} } } }$
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Answer 1

$\rm\large\underline{\text{Main idea}}$

We know that,

$\rm\cdots\longrightarrow (a^{m})^{n}=a^{mn}\ \cdots[1]$

We know that,

$\rm\cdots\longrightarrow a^{(m^{n})}=a^{m^{n}} \cdots[2]$

We know that,

$\rm\cdots\longrightarrow\sqrt[\rm n]{\rm a^{\rm m}}=a^{\frac{m}{n}}\ \cdots [3]$

We know that,

$\rm\cdots\longrightarrow(ab)^{n}=a^{n}b^{n}\ \cdots[4]$

Provided that $\rm a>0$ and $\rm b>0$.

$\rm\large\underline{\text{Explanation}}$

The required expression is the following.

$\rm\cdots\longrightarrow\sqrt{\dfrac{m^{n^{2}}n^{m^{2}}a^{(m+n)}}{(m+n)^{(m+n)^{2}}}}$

According to $\rm[2]$,

$\rm\cdots\longrightarrow\sqrt{\dfrac{m^{(n^{2})}n^{(m^{2})}a^{(m+n)}}{(m+n)^{(m+n)^{2}}}}$

According to $\rm[3]$,

$\rm\cdots\longrightarrow\left(\dfrac{m^{(n^{2})}n^{(m^{2})}a^{(m+n)}}{(m+n)^{(m+n)^{2}}}\right)^{\frac{1}{2}}$

According to $\rm[1]$ and $\rm[4]$,

$\rm\cdots\longrightarrow\dfrac{m^{\frac{n^{2}}{2}}n^{\frac{m^{2}}{2}}a^{\frac{(m+n)}{2}}}{(m+n)^{\frac{(m+n)^{2}}{2}}}$

The correct answer is choice (b) $\rm\dfrac{m^{\frac{n^{2}}{2}}n^{\frac{m^{2}}{2}}a^{\frac{m+n}{2}}}{(m+n)^{\frac{(m+n)^{2}}{2}}}$.

$\large\textrm{\underline{Extra information}}$

$\boxed{\textrm{Zero and negative exponents}}$

$\rm\cdots\longrightarrow a^{0}=1$

$\rm\cdots\longrightarrow a^{-n}=\dfrac{1}{a^{n}}$

Provided that $\rm a\neq0$.

$\boxed{\textrm{Rational exponents}}$

$\rm\cdots\longrightarrow a^{\frac{1}{n}}=\sqrt[\rm n]{\rm a}$

$\rm\cdots\longrightarrow a^{\frac{m}{n}}=\sqrt[\rm n]{\rm a^{m}}$

Provided that $\rm a>0$, for integers $\rm m$ and $\rm n(n\geq2)$.

$\boxed{\textrm{Real exponents}}$

$\rm\cdots\longrightarrow a^{m}\times a^{n}=a^{mn}$

$\rm\cdots\longrightarrow a^{m}\div a^{n}=a^{m-n}$

$\rm\cdots\longrightarrow (a^{m})^{n}=a^{mn}$

$\rm\cdots\longrightarrow (ab)^{n}=a^{n}b^{n}$

Provided that $\rm a>0$ and $\rm b>0$.

$\large\textrm{\underline{Quesion corner}}$

$\large{\red{\bigstar}\textrm{\underline{Question}}}$

If $\rm x=\sqrt[6]{2}$, simplify $\rm\dfrac{1+x+x^2+\cdots+x^{10}}{x^{-2}+x^{-3}+\cdots+x^{-12}}$.

$\large\red{\bigstar}\textrm{\underline{Answer}}$

If we take the exponent 6 on both sides,

$\rm\cdots\longrightarrow x^{6}=2$

The given expression is the following.

$\rm\cdots\longrightarrow\dfrac{x^{10}+x^{9}+x^{8}+\cdots+1}{x^{-12}(x^{10}+x^{9}+x^{8}+\cdots+1)}$

$\rm\cdots\longrightarrow\dfrac{1}{x^{-12}}$

$\rm\cdots\longrightarrow x^{12}$

$\rm\cdots\longrightarrow(x^{6})^{2}$

$\rm\cdots\longrightarrow4$

Answer 2

Step-by-step explanation:

given :

• Find the value of :
• $\displaystyle \rm \: \sqrt{ \frac{ {x}^{ {y}^{2} } {y}^{ {x}^{2} } {z}^{(x + y)} }{ {(x + y)}^{ {(x + y)}^{2} } } }$

solution :

• please check the attached file