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\sqrt[3]{ \sqrt[3]{2} - 1} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{ (1)/(9) } \\ \\ If \: a>b \: \: , \: \: Find \: \: (a+2b) \:.

User Mbrt
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2 Answers

4 votes
Define

c=\sqrt[3]{\sqrt[3]{2}-1}-\sqrt[3]{(1)/(9)} \approx 0.157435964092

Then you have the symmetrical equation

c=\sqrt[3]{a}+\sqrt[3]{b}
which can be solved for b to give

b=(c-\sqrt[3]{a})^(3)

Substituting into your expression gives

a+2b=a+2(c-\sqrt[3]{a})^(3)

The requirement that a > b means this is only relevant for

a > ((c)/(2))^(3) \approx 0.000487777605001

The attached graphs show the general shape of a+2b and some detail near the origin. "a" is plotted on the x-axis; "b" is plotted on the y-axis.
\sqrt[3]{ \sqrt[3]{2} - 1} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{ (1)/(9) } \\ \\ If-example-1
\sqrt[3]{ \sqrt[3]{2} - 1} = \sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{ (1)/(9) } \\ \\ If-example-2
User Henry Pham
by
8.1k points
6 votes
Step One
Subtract cube root 1/9 to the left hand side. Or subtract cube root (1/9) from both sides.

\sqrt[3]{ \sqrt[3]{2} -1 } - \sqrt[3]{ (1)/(9) } = \sqrt[3]{a} + \sqrt[3]{b}

Step Two.
There is a minus sign in front of
{-}\sqrt[3]{ (1)/(9) }
We must get rid of it. Because it is a minus in front of a cube root, we can bring it inside the cube root sign like so, and make it a plus out side the cube root sign

{+}\sqrt[3]{ (-1)/(9) }

Step Three
Write the Left side with the minus sign placed in the proper place

\sqrt[3]{ \sqrt[3]{2} -1 } + \sqrt[3]{ (-1)/(9) } = \sqrt[3]{a} + \sqrt[3]{b}

Step Four
Equate cube root b with cube root (-1/9)

\sqrt[3]{b} = \sqrt[3]{ (-1)/(9) }

Step Five
Equate the cube root of a with what's left over on the left

\sqrt[3]{ \sqrt[3]{2} -1 } = \sqrt[3]{a}

Step 6.
I'll just work with b for a moment.
Cube both sides of cube root (b) = cube root (-1/9)

\sqrt[3]{b} ^(3) =\sqrt[3]{ (-1)/(9) }^3}

\text{b =} (-1)/(9)

\text{2b =}(-2)/(9)

Step seven
the other part is done exactly the same way
a = cuberoot(2) - 1.

What you do from here is up to you. It is not pleasant.
Is this clearer?

a + 2b should come to cuberoot(2) - 1 - 2/9
a + 2b should come to cuberoot(2) - 11/9

I hope a person is marking this. I wonder how many of your class mates got it.
User Tssch
by
9.0k points

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