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. $x^{\frac{1}{p}+\frac{1}{q}+\frac{1}{2}}$
• B. $0$
• C. $x^{pq+qr+rp}$
• D. $1$

Answer 1

Answer: (D)

$1$

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

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

On adding all the powers of $x$, we get

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

$=x^{\frac{p(q-r)+q(r-p)+r(p-q)}{pqr}}$

$=x^{\frac{0}{pqr}}=1$