Exponents!!!

Algebra Level 3

( x 4 ) 4 x 4 = ( x 4 ) 4 x 4 \large {\left(\sqrt[4]{x}\right)}^{4x^4} = {\left(x^4\right)}^{4\sqrt[4]{x}}

Find the real value of x x that satisfies the equation above.

If the answer is of the form a × b 15 , a × \sqrt[15]{b}, then submit the value of a + b . a + b.

Note: Here, x 0 , 1 x \neq 0,1 .


This is one part of the set Fun with exponents .


The answer is 4.

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2 solutions

Ashish Menon
Apr 24, 2016

( x 4 ) 4 x 4 = ( x 4 ) 4 x 4 ( x 1 4 ) 4 x 4 = ( x 4 ) 4 x 1 4 x x 4 = x 16 x 1 4 Equating the powers : x 4 = 16 x 1 4 Raising root of 4 : x = 2 x 1 16 x x 1 16 = 2 x 1 1 16 = 2 x 15 16 = 2 Raising power of 16 : x 15 = 2 16 Raising root of 15 : x = 2 16 15 x = 2 15 × 2 15 x = 2 2 15 \begin{aligned} {\left(\sqrt[4]{x}\right)}^{4x^4} & = {\left(x^4\right)}^{4\sqrt[4]{x}}\\ {\left(x^{\tfrac{1}{4}}\right)}^{4x^4} & = {\left(x^4\right)}^{4x^{\frac{1}{4}}}\\ x^{x^4} & = x^{16x^{\frac{1}{4}}}\\ \text{Equating the powers}:-\\ x^4 & =16x {\tfrac{1}{4}}\\ \text{Raising root of 4}:-\\ x & = 2x^{\tfrac{1}{16}}\\ \dfrac{x}{x^{\tfrac{1}{16}}} & = 2\\ x^{1 - \tfrac{1}{16}} & = 2\\ x^{\tfrac{15}{16}} & = 2\\ \text{Raising power of 16}:-\\ x^{15} & = 2^{16}\\ \text{Raising root of 15}:-\\ x & = \sqrt[15]{2^{16}}\\ x & = \sqrt[15]{2^{15} × 2}\\ x & = 2\sqrt[15]{2} \end{aligned}


a + b = 2 + 2 = 4 \therefore a + b = 2 + 2 = \boxed{4}

Typo in fifth line. you have missed " ^ " before 1/4. May I suggest a few short cuts ?
1] After \ press THREE spaces and enter. So the next sentence would go to the next line.
2] when ever we have a single digit, {.} are not required. \frac {1}{2} can be written as
\frac12. But if there is a non-number keep space as follows. \sqrt2 is OK but \sqrtA is not. It should be \sqrt A or \sqrt{A}.


Niranjan Khanderia - 4 years, 10 months ago

x 4 4 x 4 = ( x 4 ) 4 x 4 = ( x 1 4 ) 4 x 4 = ( x 4 ) 4 x 1 4 x x 4 = x 16 x 1 4 Equating the powers : x 4 = 16 x 1 4 x 15 4 = 2 4 x = 2 16 15 x = 2 2 15 a + b = 2 + 2 = 4 \Large \sqrt[4]x^{4x^4} = \left(x^4\right)^{4\sqrt[4]x}\\ \Large =\left(x^{\frac 1 4}\right)^{4x^4} = {\left(x^4\right)}^{4x^{\frac1 4}}\\ \Large x^{x^4} = x^{16x^{\frac 1 4}}\\ \text{Equating the powers}:-\\ \Large x^4 =16x^ {\frac1 4}\\ \Large x^{\frac{15}{4}} =2^4 \\ \Large x=\sqrt[15]{ 2^{16}}\\ \Large x=2\sqrt[15] 2 \\ a+b=2+2=4

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