Solve (x^2(2x-4)^(1/2))^(1/2)

Question

Simplify the expression

x 4 4 x - 8

Evaluate

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

x^{2} (2 x - 4)^{\frac{1}{2}}

Write the expression as a product where the root of one of the factors can be evaluated

2 x^{2} (x - 2)^{\frac{1}{2}}

Rewrite the expression

x^{2} \times 2 (x - 2)^{\frac{1}{2}}

Calculate

x 2 (x - 2)^{\frac{1}{2}}

Solution

More Steps

Calculate

2 (x - 2)^{\frac{1}{2}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

2 x - 2

Use a = n a^{n} to transform the expression

2^{2} \times x - 2

Calculate

2^{2} (x - 2)

Calculate

4 (x - 2)

Use m n a = mn a to simplify the expression

4 4 (x - 2)

Calculate

4 4 x - 8

x 4 4 x - 8

Show Solution

x = 2

Evaluate

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}}

To find the roots of the expression,set the expression equal to 0

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} = 0

Find the domain

More Steps

Evaluate

{2 x - 4 \geq 0 x^{2} (2 x - 4)^{\frac{1}{2}} \geq 0

Calculate

More Steps

Evaluate

2 x - 4 \geq 0

Move the constant to the right side

2 x \geq 0 + 4

Removing 0 doesn't change the value,so remove it from the expression

2 x \geq 4

Divide both sides

\frac{2 x}{2} \geq \frac{4}{2}

Divide the numbers

x \geq \frac{4}{2}

Divide the numbers

x \geq 2

{x \geq 2 x^{2} (2 x - 4)^{\frac{1}{2}} \geq 0

Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x

{x \geq 2 x \in R

Find the intersection

x \geq 2

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} = 0, x \geq 2

Calculate

(x^{2} (2 x - 4)^{\frac{1}{2}})^{\frac{1}{2}} = 0

The only way a root could be 0 is when the radicand equals 0

x^{2} (2 x - 4)^{\frac{1}{2}} = 0

Separate the equation into 2 possible cases

x^{2} = 0 (2 x - 4)^{\frac{1}{2}} = 0

The only way a power can be 0 is when the base equals 0

x = 0 (2 x - 4)^{\frac{1}{2}} = 0

Solve the equation

More Steps

Evaluate

(2 x - 4)^{\frac{1}{2}} = 0

The only way a root could be 0 is when the radicand equals 0

2 x - 4 = 0

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

2 x = 0 + 4

Removing 0 doesn't change the value,so remove it from the expression

2 x = 4

Divide both sides

\frac{2 x}{2} = \frac{4}{2}

Divide the numbers

x = \frac{4}{2}

Divide the numbers

More Steps

Evaluate

\frac{4}{2}

Reduce the numbers

\frac{2}{1}

Calculate

2

x = 2

x = 0 x = 2

Check if the solution is in the defined range

x = 0 x = 2, x \geq 2

Solution

x = 2

Show Solution