Question

I’m done with it all but imma keep my promise and give 100 to the person who can solve this one for my challenge question on here

Answer 1

Answer:

$\boxed{{a}^{4} + {b}^{4} + {c}^{4} =50}$

Step-by-step explanation:

$a+b+c = 4$

Squaring both the side,

${(\underline{a +b} + c)}^{2} = {4}^{2}$

${(a + b)}^{2} + {c}^{2} + 2(a + b)c \: = 16$

${a}^{2} + 2ab+ {b}^{2} + {c}^{2} + 2ac + 2bc = 16$

${a}^{2} + {b}^{2} + {c}^{2} + 2ab+ 2ac + 2bc = 16$

$10 + 2ab+ 2ac + 2bc = 16$

$2ab+ 2ac + 2bc = 16 - 10 = 6$

$2(ab + bc + ac) = 6$

$\fbox{ab + bc + ac = 3}$

Now we know that,

${a}^{3} + {b}^{3} + {c}^{3} -3abc = ( a + b + c)( {a}^{2} + {b}^{2} + {c}^{2} - ab - bc - ca)$

$22 -3abc = ( 4)( 10 - (ab + bc + ca))$

$22 -3abc = ( 4)( 10 - 3)$

$3abc = 22 - 40 + 12$

$abc = \frac{ - 6}{3} = - 2$

Now, we know that

$ab + bc + ac = 3$

Squaring both the side,

${(ab + bc + ac)}^{2} = {3}^{2}$

${(ab)}^{2} + {(ac)}^{2} + {(bc)}^{2} + 2 (abbc + abac + bcac) = 9$

${(ab)}^{2} + {(ac)}^{2} + {(bc)}^{2} + 2 (a + b + c)(abc) = 9$

${(ab)}^{2} + {(ac)}^{2} + {(bc)}^{2} + 2 \times 4 \times ( - 2) = 9$

${(ab)}^{2} + {(ac)}^{2} + {(bc)}^{2} - 16= 9$

${(ab)}^{2} + {(ac)}^{2} + {(bc)}^{2} = 25$

Now we are given that,

${a}^{2} + {b}^{2} + {c}^{2} = 10$

Squaring both the side,

${({a}^{2} + {b}^{2} + {c}^{2})}^{2} = {10}^{2}$

${ {a}^{2} }^{2} + { {b}^{2} }^{2} + { {c}^{2} }^{2} + 2( {a}^{2} {b}^{2} + {b}^{2} {c}^{2} + {a}^{2} {c}^{2} ) = 100$

${a}^{4} + {b}^{4} + {c}^{4} + 2( {a}^{2} {b}^{2} + {b}^{2} {c}^{2} + {a}^{2} {c}^{2} ) = 100$

${a}^{4} + {b}^{4} + {c}^{4} + 2( 25 ) = 100$

${a}^{4} + {b}^{4} + {c}^{4} + 50= 100$

$\boxed{{a}^{4} + {b}^{4} + {c}^{4} =50}$

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Answer 2

There is a secret rule hiding inbetween these equations.If we can spot out the rule we won't have to do lengthy calculations .we can do it in one line even then .

First note that degree of equation is taken as d for all the context below provided

When

d=1

• The expression yields 4

d=2

• expression yeilds=10

d=3

• expression yields=22

d=4 we need what the expression yields

See if we observe

For d=1

• The expression equals =4

Next for d=2 and d=3

The respective yields are 10 and 22

Spot the secret

• 4²-6=10
• 4²+6=22

70% secret exposed,rest we do

Find the difference

• 22-10=12

Then

• 6 is 4×1+2

Next one must be 4×12+2 (As it's difference before)

It's exactly

• 48+2=50

So the next chain for d=4 and d=5 are

• (d=2)²±50
• 10²±50
• 50 and 150

So the respective values are

• a⁴+b⁴+c⁴=50

• a⁵+b⁵+c⁵=150