Question

Consider this equation

Answer 1

$\sf \: (4x) ^{ \frac{1}{3} } - x = 0$

To raise a product to a power, raise each factor to that power

$\sf \: {4}^{ \frac{1}{3} } \times {x}^{ \frac{1}{3} } - x = 0$

Use ${a}^{ \frac{m}{n} } = \sqrt[n]{ {a}^{m} }$ to transform the expression

$\sf \sqrt[3]{4} \sqrt[3]{x} - x = 0$

The product of roots with the same index is equal to the root of the product

$\sf \sqrt[3]{4x} - x = 0$

Move the variable to the right-hand side and change its sign

$\sf \sqrt[3]{4x} = x$

Raise both sides of the equation to the power of 3

$\sf {4x} = {x}^{3}$

Move the variable to the left-hand side and change its sign

$\sf4x - {x}^{3} = 0$

Factor out x from the expression

$\sf \: x(4x - {x}^{2} ) = 0$

When the product of factors equals 0, at least one factor is 0

$\sf \: x = 0 \\ \sf \: 4 - {x}^{2} = 0$

Solve the equation for x

$\boxed{ \tt x_{1} = 0, x_{2} = 2}$

Checking the solution shows that -2 as x is not a solution to the given equation