# Indices (Equation)

(CIE 0606/2020/w/11/Q3)
(a) Write $\dfrac{\sqrt p(qr^2)^{\frac 13}}{(q^3p)^{-1}r^3}$ in the form $p^aq^br^c$, where $a, b$ and $c$ are constants. [3]

(b) Solve $6x^{\frac 23}-5x^{\frac 13}+1=0$.

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*********math solution*************

(a) $\begin{array}[t]{rcll}\dfrac{p^{\frac 12}(qr^2)^{\frac 13}}{(q^3p)^{-1}r^3}&=&p^{\frac 12}q^{\frac 13}r^{\frac 23}(q^3p)r^{-3}\\&=&p^{\frac 12+1}q^{\frac 13+3}r^{\frac 23-3}\\ &=&p^{\frac 32}q^{\frac{10}{3}}r^{\frac 73}\end{array}$

$\def\y{x^{\frac{1}{3}}}$

(b) $\begin{array}[t]{rcll}6x^{\frac 23}-5\y+1&=&0\\6(\y)^2-5(\y)+1&=&0\\ (3\y-1)(2\y-1)&=&0\\ 3\y=1 &\mbox{(or)}&2\y=1\\ \y=\dfrac 13&\mbox{(or)}& \y=\dfrac 12\\ x=\dfrac{1}{27} &\mbox{(or)}& x=\dfrac 18\end{array}$

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