Solve [(1 k)^(1/k) - 1] [(1 k)^(1/k) - 1]

Question

Simplify the expression

k^{\frac{2}{k}} - 2 k^{\frac{1}{k}} + 1

Evaluate

((1 \times k)^{\frac{1}{k}} - 1) ((1 \times k)^{\frac{1}{k}} - 1)

Any expression multiplied by 1 remains the same

(k^{\frac{1}{k}} - 1) ((1 \times k)^{\frac{1}{k}} - 1)

Any expression multiplied by 1 remains the same

(k^{\frac{1}{k}} - 1) (k^{\frac{1}{k}} - 1)

Use the the distributive property to expand the expression

k^{\frac{1}{k}} \times k^{\frac{1}{k}} + k^{\frac{1}{k}} (- 1) - k^{\frac{1}{k}} - (- 1)

Multiply the terms

More Steps

Evaluate

k^{\frac{1}{k}} \times k^{\frac{1}{k}}

Multiply the terms with the same base by adding their exponents

k^{\frac{1}{k} + \frac{1}{k}}

Calculate

k^{\frac{2}{k}}

k^{\frac{2}{k}} + k^{\frac{1}{k}} (- 1) - k^{\frac{1}{k}} - (- 1)

Multiply the terms

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

When there is - in front of an expression in parentheses change the sign of each term of the expression and remove the parentheses

k^{\frac{2}{k}} - k^{\frac{1}{k}} - k^{\frac{1}{k}} + 1

Solution

k^{\frac{2}{k}} - 2 k^{\frac{1}{k}} + 1

Show Solution

k = 1

Evaluate

((1 \times k)^{\frac{1}{k}} - 1) ((1 \times k)^{\frac{1}{k}} - 1)

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

((1 \times k)^{\frac{1}{k}} - 1) ((1 \times k)^{\frac{1}{k}} - 1) = 0

Find the domain

More Steps

Evaluate

{k \neq = 0 1 \times k > 0

Any expression multiplied by 1 remains the same

{k \neq = 0 k > 0

Find the intersection

k > 0

((1 \times k)^{\frac{1}{k}} - 1) ((1 \times k)^{\frac{1}{k}} - 1) = 0, k > 0

Calculate

((1 \times k)^{\frac{1}{k}} - 1) ((1 \times k)^{\frac{1}{k}} - 1) = 0

Any expression multiplied by 1 remains the same

(k^{\frac{1}{k}} - 1) ((1 \times k)^{\frac{1}{k}} - 1) = 0

Any expression multiplied by 1 remains the same

(k^{\frac{1}{k}} - 1) (k^{\frac{1}{k}} - 1) = 0

Calculate

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

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

k^{\frac{1}{k}} - 1 = 0

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

k^{\frac{1}{k}} = 0 + 1

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

k^{\frac{1}{k}} = 1

This equation equals 1 only if the base equals 1 or -1 or the exponent equals 0

k = 1 k = - 1 \frac{1}{k} = 0

Evaluate

More Steps

Evaluate

\frac{1}{k} = 0

Cross multiply

1 = k \times 0

Simplify the equation

1 = 0

The statement is false for any value of k

k \in \emptyset

k = 1 k = - 1 k \in \emptyset

Evaluate

k = 1 k = - 1

Check if the solution is in the defined range

k = 1 k = - 1, k > 0

Solution

k = 1

Show Solution