QuestionRationalize the denominator of $\frac{4}{\sqrt{x}+1}$ .

Studdy Solution

STEP 1

Assumptions1. We need to rationalize the denominator of the given expression. . The given expression is $\frac{4}{\sqrt{x}+1}$ .
3. To rationalize means to eliminate the square root from the denominator.

STEP 2

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{x}+1$ is $\sqrt{x}-1$ .
$\frac{4}{\sqrt{x}+1} \times \frac{\sqrt{x}-1}{\sqrt{x}-1}$

STEP 3

Multiply the numerators together and the denominators together.
$\frac{(\sqrt{x}-1)}{(\sqrt{x}+1)(\sqrt{x}-1)}$

STEP 4

The denominator is a difference of squares, which can be simplified to $a^2 - b^2$ , where $a = \sqrt{x}$ and $b =1$ .
$\frac{4(\sqrt{x}-1)}{x -1^2}$

STEP 5

implify the denominator.
$\frac{4(\sqrt{x}-1)}{x -1}$

STEP 6

istribute the numerator.
$\frac{4\sqrt{x} -4}{x -1}$ This is the rationalized form of the given expression.

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QuestionRationalize the denominator of $\frac{4}{\sqrt{x}+1}$ .

Studdy Solution

STEP 1

Assumptions1. We need to rationalize the denominator of the given expression. . The given expression is $\frac{4}{\sqrt{x}+1}$ .
3. To rationalize means to eliminate the square root from the denominator.

STEP 2

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{x}+1$ is $\sqrt{x}-1$ .
$\frac{4}{\sqrt{x}+1} \times \frac{\sqrt{x}-1}{\sqrt{x}-1}$

STEP 3

Multiply the numerators together and the denominators together.
$\frac{(\sqrt{x}-1)}{(\sqrt{x}+1)(\sqrt{x}-1)}$

STEP 4

The denominator is a difference of squares, which can be simplified to $a^2 - b^2$ , where $a = \sqrt{x}$ and $b =1$ .
$\frac{4(\sqrt{x}-1)}{x -1^2}$

STEP 5

implify the denominator.
$\frac{4(\sqrt{x}-1)}{x -1}$

STEP 6

istribute the numerator.
$\frac{4\sqrt{x} -4}{x -1}$ This is the rationalized form of the given expression.

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QuestionRationalize the denominator of 4x+1\frac{4}{\sqrt{x}+1}x​+14​.

Studdy Solution

STEP 1

Assumptions1. We need to rationalize the denominator of the given expression. . The given expression is 4x+1\frac{4}{\sqrt{x}+1}x​+14​.3. To rationalize means to eliminate the square root from the denominator.

STEP 2

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of x+1\sqrt{x}+1x​+1 is x−1\sqrt{x}-1x​−1.4x+1×x−1x−1\frac{4}{\sqrt{x}+1} \times \frac{\sqrt{x}-1}{\sqrt{x}-1} x​+14​×x​−1x​−1​

STEP 3

Multiply the numerators together and the denominators together.(x−1)(x+1)(x−1)\frac{(\sqrt{x}-1)}{(\sqrt{x}+1)(\sqrt{x}-1)} (x​+1)(x​−1)(x​−1)​

STEP 4

The denominator is a difference of squares, which can be simplified to a2−b2a^2 - b^2a2−b2, where a=xa = \sqrt{x}a=x​ and b=1b =1b=1.4(x−1)x−12\frac{4(\sqrt{x}-1)}{x -1^2} x−124(x​−1)​

STEP 5

implify the denominator.4(x−1)x−1\frac{4(\sqrt{x}-1)}{x -1} x−14(x​−1)​

STEP 6

istribute the numerator.4x−4x−1\frac{4\sqrt{x} -4}{x -1} x−14x​−4​This is the rationalized form of the given expression.

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QuestionRationalize the denominator of $\frac{4}{\sqrt{x}+1}$ .

Assumptions1. We need to rationalize the denominator of the given expression. . The given expression is $\frac{4}{\sqrt{x}+1}$ .
3. To rationalize means to eliminate the square root from the denominator.

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{x}+1$ is $\sqrt{x}-1$ .
$\frac{4}{\sqrt{x}+1} \times \frac{\sqrt{x}-1}{\sqrt{x}-1}$

Multiply the numerators together and the denominators together.
$\frac{(\sqrt{x}-1)}{(\sqrt{x}+1)(\sqrt{x}-1)}$

The denominator is a difference of squares, which can be simplified to $a^2 - b^2$ , where $a = \sqrt{x}$ and $b =1$ .
$\frac{4(\sqrt{x}-1)}{x -1^2}$

implify the denominator.
$\frac{4(\sqrt{x}-1)}{x -1}$

istribute the numerator.
$\frac{4\sqrt{x} -4}{x -1}$ This is the rationalized form of the given expression.