# Number system practice

Rationalise & find value of cubic expression

#### Question: If x = $$frac{1}{8-\sqrt{60}}\\$, what is the value of $x^{3}-5x^{2}+8x-4?\\$ #### Video Explanation #### Explanatory Answer In all these questions, before plugging the value of x in the expression whose value is to be computed, rationalize x and get x expressed with a rational denominator. You are likely to find a clue after this step. ##### Step 1: Rationalise $\frac{1}{8 - \sqrt{60}}\\$ $\frac{1}{8 - \sqrt{60}}\\$ x $\frac{8 + \sqrt{60}}{8 + \sqrt{60}}\\$ =$\frac{8 + \sqrt{60}}{4}\\$ = 2 + $\frac{\sqrt{15}}{2}\\$ Or x - 2 = $\frac{\sqrt{15}}{2}\\$ ##### Step 2: Rewrite $x^3 - 5x^2 + 8x - 4\\$ Because x = 2 + $\frac{\sqrt{15}}{2}\\$, try and express $x^3 - 5x^2 + 8x - 4\\$ in terms of ${$x-2$}^3, {(x-2)}^2$ and $x - 2\\$. ${$x - 2$}^3 = x^3 - 6x^2 + 12x - 8$ --------$1)
${$x - 2)}^2 = x^2 - 4x + 4$ --------$2)

Add or subtract the two equations to check whether you get the given expression

(1) + (2) = $x^3 - 5x^2 + 8x - 4$\$ So, $x^3 - 5x^2 + 8x - 4\\$ = ${$x-2$}^3$ + ${$x-2$}^2$ =${$$frac{\sqrt{15}}{2}$^3}$\$ + ${$$frac{\sqrt{15}}{2}$^2}$\$ =$\frac{15\sqrt{15}}{8}\\$ + $\frac{15}{4}\\$ =$\frac{15{$\sqrt{15}+2$}}{8}$\$