A easy one.

Algebra Level 3

If ${ 4 }^{ x }+{ 4 }^{ -x }=7$ , then what is ${ 8 }^{ x }+{ 8 }^{ -x }$ equals ?

2 solutions

Chaitnya Shrivastava
Feb 20, 2016

Good clear explanation.

Vishwak Srinivasan
Jul 15, 2015

$4^x + 4^{-x} = 7$

$\Rightarrow (4^x + 4^{-x})^3 = 343$

$\Rightarrow 64^x + 64^{-x} + 3(4^x)(4^{-x})(4^x + 4^{-x}) = 343$

$\Rightarrow 64^x + 64^{-x} + 3(7) = 343 \Rightarrow 64^x + 64^{-x} = 322$

$\Rightarrow 64^x + 64^{-x} + 2(8^x)(8^{-x}) = 322 + 2 = 324$

$\Rightarrow (8^x + 8^{-x})^2 = 18^2 \Rightarrow 8^x + 8^{-x} = \boxed{18}$

Negative solution is not allowed since $8^x + 8^{-x} \geq 2$