HW

$\large \sqrt[3]{x - 4} -\large\sqrt[3]{x-3}=-x$

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

$\sqrt [ 3 ]{ x-4 } -\sqrt [ 3 ]{ x-3 } =-x\\ \Rightarrow \left( \sqrt [ 3 ]{ x-4 } -\sqrt [ 3 ]{ x-3 } \right) ^{ 3 }=(-x)^{ 3 }\\ \Rightarrow \left( \sqrt [ 3 ]{ x-4 } \right) ^{ 3 }-\left( \sqrt [ 3 ]{ x-3 } \right) ^{ 3 }-3\sqrt [ 3 ]{ x-4 } \sqrt [ 3 ]{ x-3 } (\sqrt [ 3 ]{ x-4 } -\sqrt [ 3 ]{ x-3 } )=-x^{ 3 }\\ \Rightarrow x-4-x+3-3\sqrt [ 3 ]{ x^{ 2 }-7x+12 } (-x)=-x^{ 3 }\\ \Rightarrow 3x\cdot \sqrt [ 3 ]{ x^{ 2 }-7x+12 } =1-x^{ 3 }\\ \Rightarrow 27x^{ 3 }(x^{ 2 }-7x+12)=1-3x^{ 3 }+3x^{ 6 }-x^{ 9 }\\ \Rightarrow x^{ 9 }-3x^{ 6 }+27x^{ 5 }-189x^{ 4 }+327x^{ 3 }-1$

Then apply Newton Raphson Method

Md Mehedi Hasan - 3 years, 6 months ago

Rewrite the equation with $\sqrt[3]{x - 4} = y$ and $x = y^3 + 4$
Solve for $y$
Put the value of $y$ in $\sqrt[3]{x - 4} = y$ .
Solve for $x$

Munem Shahriar - 3 years, 6 months ago

When I rewrited it I got y^9 What should I do?

Maxim Kasnedelchev - 3 years, 6 months ago

Here is the rewritten equation:

$y - \sqrt[3]{(y^3+4) - 3} = -(y^3 + 4)$

$\Rightarrow -\sqrt[3]{y^3+4-3} = -y^3-4-y$

Cube on both sides:

$\Rightarrow (-\sqrt[3]{(y^3+4-3)})^3 = (-y^3-4-y)^3$

$\Rightarrow -y^3-1 = -y^9 -3y^7-12y^6-3y^5-24y^4-49y^3-12y^2-48y-64$

$y \approx -1.5673....$

Now solve for $x$

Munem Shahriar - 3 years, 6 months ago

Cube both sides.

Nazmus sakib - 3 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$