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avatar+738 

If \((4-a)^3 = 32\), then what is \(a\)?

 

Answer is \(a = 4-2\sqrt[3]{4}\).

 

I just need work finding the solution.

 Jul 24, 2019
 #1
avatar+8724 
+5

\((4-a)^3\ =\ 32\)

                                         Take the cube root of both sides of the equation.

\(\sqrt[3]{(4-a)^3}\ =\ \sqrt[3]{32}\)

                                         Simplify the left side with the rule  \(\sqrt[3]{n^3}\ =\ n\)

\(4-a\ =\ \sqrt[3]{32}\)

                                                    We can rewrite  32  like this because  32 = 2 * 2 * 2 * 2 * 2

\(4-a\ =\ \sqrt[3]{2\cdot2\cdot2\cdot2\cdot2}\)

                                                    We can rewrite the right side again like this...

\(4-a\ =\ \sqrt[3]{2\cdot2\cdot2}\cdot\sqrt[3]{2\cdot2}\)

                                                    And   2 * 2 * 2  =  23   and   2 * 2  =  4

\(4-a\ =\ \sqrt[3]{2^3}\,\cdot\,\sqrt[3]{4}\)

                                                    Simplify  \(\sqrt[3]{2^3}\)  again with the rule  \(\sqrt[3]{n^3}\ =\ n\)

\(4-a\ =\ 2\,\cdot\,\sqrt[3]{4}\)

 

\(4-a\ =\ 2\sqrt[3]{4}\)

                               Add  a  to both sides of the equation.

\(4\ =\ 2\sqrt[3]{4}+a\)

                               Subtract  \(2\sqrt[3]{4}\)  from both sides of the equation.

\(4-2\sqrt[3]{4}\ =\ a\)

 

\(a\ =\ 4-2\sqrt[3]{4}\)-

 Jul 24, 2019
 #2
avatar+1014 
+1

Nice! 

Nickolas  Jul 24, 2019
 #3
avatar+18961 
+2

(4-a)^3 = 32      cube root both sides   (note that 32 = 2^5 )

 

(4-a) = cubrt(2^5)      Simplify the right side

(4-a) = 2 cubrt(4)      Add 'a' to both sides and subtract 2 cubrt(4) from both sides

4- 2 cubrt(4) = a

 Jul 24, 2019

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