Question

The value of ${\left(\frac{x^q}{x^r}\right)}^{\frac{1}{qr}}\times {\left(\frac{x^r}{x^p}\right)}^{\frac{1}{rp}}\times {\left(\frac{x^p}{x^q}\right)}^{\frac{1}{pq}}$ is equal to

• A. $x^{\frac{1}{p}+\frac{1}{q}+\frac{1}{r}}$
• B. $0$
• C. $x^{pq+qr+rp}$
• D. $1$

Answer 1

Answer: (D)

$1$

$\Rightarrow {\left(\frac{x^q}{x^r}\right)}^{\frac{1}{qr}}\times {\left(\frac{x^r}{x^p}\right)}^{\frac{1}{rp}}\times {\left(\frac{x^p}{x^q}\right)}^{\frac{1}{pq}}$

$\Rightarrow (x^{q-r}{)}^{\frac{1}{qr}}\times (x^{r-p}{)}^{\frac{1}{rp}}\times (x^{p-q}{)}^{\frac{1}{pq}}$

$\Rightarrow x^{\frac{q-r}{qr}}\times x^{\frac{r-p}{rp}}\times x^{\frac{p-q}{pq}}$

$\Rightarrow x^{\frac{1}{r}-\frac{1}{q}}\times x^{\frac{1}{p}-\frac{1}{r}}\times x^{\frac{1}{q}-\frac{1}{p}}$

$\Rightarrow x^{\frac{1}{r}-\frac{1}{q}+\frac{1}{p}-\frac{1}{r}+\frac{1}{q}-\frac{1}{p}}$

$\Rightarrow x^0$

$\Rightarrow 1$