What is the value of $| m |$ $|m|$ in $\sqrt[3]{m + 9} = 3 + \sqrt[3]{m - 9}$ $root(3)(m+9)=3+root(3)(m-9)$ ?

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Answer 1 · 2016-10-02T01:28:13

$| m | = 4 \sqrt{5}$ $|m| = 4sqrt(5)$

We will make use of the expansion of the cube of a binomial

${(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ $(a+b)^3 = a^3+3a^2b+3ab^2+b^3$

$a x^{2} + b x + c = 0 \Rightarrow x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $ax^2+bx+c=0 => x = (-b+-sqrt(b^2-4ac))/(2a)$

Proceeding,

$\sqrt[3]{m + 9} = 3 + \sqrt[3]{m - 9}$ $root(3)(m+9) = 3 + root(3)(m-9)$

$\Rightarrow m + 9 = {(3 + \sqrt[3]{m - 9})}^{3}$ $=> m+9 = (3+root(3)(m-9))^3$

$\Rightarrow m + 9 = 27 + 27 \sqrt[3]{m - 9} + 9 {(\sqrt[3]{m - 9})}^{2} + m - 9$ $=> m+9 = 27 + 27root(3)(m-9) + 9(root(3)(m-9))^2+m-9$

$\Rightarrow 9 {(\sqrt[3]{m - 9})}^{2} + 27 \sqrt[3]{m - 9} + 9 = 0$ $=> 9(root(3)(m-9))^2 + 27root(3)(m-9) + 9 = 0$

$\Rightarrow {(\sqrt[3]{m - 9})}^{2} + 3 \sqrt[3]{m - 9} + 1 = 0$ $=> (root(3)(m-9))^2 + 3root(3)(m-9) + 1 = 0$

$\Rightarrow \sqrt[3]{m - 9} = \frac{- 3 \pm \sqrt{{(- 3)}^{2} - 4 (1) (1)}}{2 (1)}$ $=> root(3)(m-9) = (-3+-sqrt((-3)^2-4(1)(1)))/(2(1))$

$\Rightarrow \sqrt[3]{m - 9} = \frac{- 3 \pm \sqrt{5}}{2}$ $=> root(3)(m-9) = (-3+-sqrt(5))/2$

$\Rightarrow 2 \sqrt[3]{m - 9} = - 3 \pm \sqrt{5}$ $=> 2root(3)(m-9) = -3 +-sqrt(5)$

$\Rightarrow 8 (m - 9) = {(- 3 \pm \sqrt{5})}^{3}$ $=> 8(m-9) = (-3 +- sqrt(5))^3$

$\Rightarrow 8 m - 72 = - 27 \pm 27 \sqrt{5} - 45 \pm 15 \sqrt{5}$ $=> 8m - 72 = -27 +- 27sqrt(5) - 45 +- 15sqrt(5)$

$\Rightarrow 8 m - 72 = - 72 \pm 32 \sqrt{5}$ $=> 8m - 72 = -72 +- 32sqrt(5)$

$\Rightarrow 8 m = \pm 32 \sqrt{5}$ $=> 8m = +-32sqrt(5)$

$\Rightarrow m = \pm 4 \sqrt{5}$ $=> m = +-4sqrt(5)$

$:. |m| = 4sqrt(5)$

What is the value of |m||m||m| in root(3)(m+9)=3+root(3)(m-9)3√m+9=3+3√m−9root(3)(m+9)=3+root(3)(m-9) ?