Question

The cube of number is 8 times the cube of another number. If the sum of the cubes of the numbers is 243, then find the difference of the number​

Answer 1

AnswEr

Let's consider that two numbers are x & y.

⠀⠀⠀ $\underline{\boldsymbol{According\: to \:the\: Question :}}$⠀⠀⠀

⠀⠀⠀Cube of number is 8 times the cube of another number :

$:\implies\sf x^3 = 8y^3$⠀⠀⠀⠀⠀ -eq(1).

Sum of the cubes of the numbers is 243.

$:\implies\sf x^3 + y^3 = 243$ ⠀⠀ -eq(2).

⠀⠀⠀From both Equations :

$:\implies\sf 8y^3 + \cancel{ y^3} = 243 \\\\\\:\implies\sf x^3 +\cancel{ y^3} = 243 \\\\\\:\implies\sf \purple{9y^3 = 243}\\\\\\:\implies\sf y^3 = \dfrac{\cancel{243}}{\cancel{\: 9}} \\\\\\:\implies\sf y^3 = 27 \\\\\\:\implies\boxed{\frak{\pink{y = 3}}}$

⠀⠀

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀Substituting value of y in eq (1).

$:\implies\sf x^3 = 8y^3 \\\\\\:\implies\sf x^3 = 8(3)^3\\\\\\:\implies\sf x^3 = 8(27) \\\\\\:\implies\sf x^3 = 216\\\\\\:\implies\boxed{\frak{\pink{x = 6}}}$

⠀$\therefore$ Value of y & x is 3 & 6.

⠀$\bigstar$ Difference of the numbers :

$:\implies\sf Numbers = x - y \\\\\\:\implies\sf 6 - 3\\\\\\:\implies\boxed{\frak{\purple{3}}}$

⠀⠀Hence, Difference of the numbers is 3.

Answer 2

Answer:

$\huge{\bold{Heya \: dost \: ! }}$

$x {}^{3} = y {}^{3} ..........(1)$

$x {}^{3} + y {}^{3} = 243 \: ............(2)$

$xy {}^{3} + y {}^{3} = 243$

$9y {}^{3} = 243$

$y {}^{3} = 27$

$y = 3..............(3)$

$x {}^{3} = 8 \times 27.........from(1) \: and \: (3)$

$x = 2 \times 3 = 6.......Taking \: cube \: roots \: on \: both \: sides$

$Difference \: = 6 - 3 = 3$