Find the value

Question:

Find the value

i. $\frac{173 \times 173 \times 173+127 \times 127 \times 127}{173 \times 173-173 \times 127+127 \times 127}$

ii. $\frac{1.2 \times 1.2 \times 1.2-0.2 \times 0.2 \times 0.2}{1.2 \times 1.2+1.2 \times 0.2+0.2 \times 0.2}$

iii. $\frac{155 \times 155 \times 155-55 \times 55 \times 55}{155 \times 155+155 \times 55+55 \times 55}$

Solution:

i. $\frac{173 \times 173 \times 173+127 \times 127 \times 127}{173 \times 173-173 \times 127+127 \times 127}$

$=\frac{173^{3}+127^{3}}{173^{2}-173 \times 127+1272}$

$=\frac{(173+127)\left(173^{2}-173 \times 127+127^{2}\right)}{173^{2}-173 \times 127+127^{2}}$

$\therefore\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right]$

= (173 + 127)

= 300

ii. $\frac{1.2 \times 1.2 \times 1.2-0.2 \times 0.2 \times 0.2}{1.2 \times 1.2+1.2 \times 0.2+0.2 \times 0.2}$

$=\frac{1.2^{3}-0.23}{1.2^{2}+1.2 \times 0.2+0.2^{2}}$

$=\frac{(1.2-0.2)\left((1.2)^{2}+1.2 \times 0.2+(0.2)^{2}\right)}{1.2^{2}+1.2 \times 0.2+0.2^{2}}$

$\left[\therefore a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$

= (1.2 − 0.2)

= 1.0

iii. $\frac{155 \times 155 \times 155-55 \times 55 \times 55}{155 \times 155+155 \times 55+55 \times 55}$

$=\frac{155^{3}-55^{3}}{155^{2}+155 \times 55+55^{2}}$

$=\frac{(155-55)\left(155^{2}+155 \times 55+55^{2}\right)}{155^{2}+155 \times 55+55^{2}}$

$\left[\therefore a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$

= (155 - 55)

= 100

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