CAIE IGCSE Additional Math 4.3 substitution for related equations

Solves or factorises $x^{\frac{1}{3}}-x^{\frac{1}{6}}-2=0$ oe:
$$\left(x^{\frac{1}{6}}+1\right)\left(x^{\frac{1}{6}}-2\right)$$

or when substituting $y=x^{\frac{1}{3}}$ factorises to
$$\left(y^{\frac{1}{2}}+1\right)\left(y^{\frac{1}{2}}-2\right)$$

M1

A substitution may be used
$$\left(x^{\frac{1}{3}}\right)^{6}-\left(x^{\frac{1}{6}}\right)^{6}=2^{6} \text { scores } 0 \text { marks }$$$\left(x^{\frac{1}{3}}\right)-\left(x^{\frac{1}{6}}\right)=x^{\log _{x} 2}$ then $\frac{1}{3}-\frac{1}{6}=\log _{x} 2$ scores 0 marks

$\left[x^{\frac{1}{6}}=-1\right], \quad x^{\frac{1}{6}}=2$

A1

x=64[x=1]

A1

Rejects $x^{\frac{1}{6}}=-1$ or x=1 ignored at some stage,
soi

B1