Exponents Everywhere!

Algebra Level 4

$\Large {x^{\sqrt [3]{x^{2}}}=\left (\sqrt {x}\right)^{x}}$

Find the sum of roots of the equation above.

The answer is 9.

1 solution

Akshat Sharda
Oct 21, 2015

$\text{We can just look and say that } x=1 \text{ satisfies the equation.} \\ x^{\sqrt [3]{x^{2}}}=\left (\sqrt {x}\right)^{x}\Rightarrow x^{x^{\frac{2}{3}}}=x^{\frac{x}{2}} \\ x^{\frac{2}{3}}=\frac{x}{2} \Rightarrow x^{2}=\frac{x^{3}}{8}\Rightarrow x=8 \\ \text{Sum of values of } x=\boxed {9}$

While cancelling, you have eliminated one root i.e. $x = 0$ .
Though its an extraneous root, it will not affect the answer but you should show it in your calculations.

Akhil Bansal - 5 years, 7 months ago

0 can't be a root to this equation, because 0^0 is undefined

Doug Rauber - 5 years, 7 months ago

That's why it is an extraneous root!

shaurya gupta - 5 years, 7 months ago

We can say that the 1 will be a root by observation easily. But why it doesn't appear in the calculation process.

HIMANSHU PRAJAPATI - 5 years, 7 months ago

When we equal powers of the exponent we are assuming that x!=1 otherwise both LHS and RHS are equal to 1. So the two subcases give the complete set of roots.

vikas patidar - 5 years, 7 months ago

Did the same way.

Niranjan Khanderia - 3 years, 9 months ago

Exponents Everywhere!

The answer is 9.

1 solution

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