Math / Algebra

QuestionWhich is the rationalized form of the expression $\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7}}$ ? a. $\frac{x-\sqrt{7 x}}{x-7}$ c. $\frac{x+\sqrt{7 x}}{x-7}$ b. $\frac{\sqrt{7 x}}{7}$ d. $\frac{x}{x+7}$

Studdy Solution

STEP 1

1. We are given an expression with a radical in the denominator.
2. The goal is to rationalize the denominator.
3. Rationalizing involves eliminating the square root from the denominator.

STEP 2

1. Multiply the numerator and the denominator by the conjugate of the denominator.
2. Simplify the resulting expression.

STEP 3

Identify the conjugate of the denominator $\sqrt{x} + \sqrt{7}$ . The conjugate is $\sqrt{x} - \sqrt{7}$ .
Multiply both the numerator and the denominator by this conjugate:
$\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7}} \times \frac{\sqrt{x} - \sqrt{7}}{\sqrt{x} - \sqrt{7}}$

STEP 4

Simplify the numerator:
$\sqrt{x} \times (\sqrt{x} - \sqrt{7}) = x - \sqrt{7x}$

STEP 5

Simplify the denominator using the difference of squares formula:
$(\sqrt{x} + \sqrt{7})(\sqrt{x} - \sqrt{7}) = (\sqrt{x})^2 - (\sqrt{7})^2 = x - 7$

STEP 6

Combine the simplified numerator and denominator:
$\frac{x - \sqrt{7x}}{x - 7}$
This is the rationalized form of the expression.
The rationalized form of the expression is:
$\boxed{\frac{x - \sqrt{7x}}{x - 7}}$
This corresponds to option a.

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QuestionWhich is the rationalized form of the expression $\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7}}$ ? a. $\frac{x-\sqrt{7 x}}{x-7}$ c. $\frac{x+\sqrt{7 x}}{x-7}$ b. $\frac{\sqrt{7 x}}{7}$ d. $\frac{x}{x+7}$

Studdy Solution

STEP 1

1. We are given an expression with a radical in the denominator.
2. The goal is to rationalize the denominator.
3. Rationalizing involves eliminating the square root from the denominator.

STEP 2

1. Multiply the numerator and the denominator by the conjugate of the denominator.
2. Simplify the resulting expression.

STEP 3

Identify the conjugate of the denominator $\sqrt{x} + \sqrt{7}$ . The conjugate is $\sqrt{x} - \sqrt{7}$ .
Multiply both the numerator and the denominator by this conjugate:
$\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7}} \times \frac{\sqrt{x} - \sqrt{7}}{\sqrt{x} - \sqrt{7}}$

STEP 4

Simplify the numerator:
$\sqrt{x} \times (\sqrt{x} - \sqrt{7}) = x - \sqrt{7x}$

STEP 5

Simplify the denominator using the difference of squares formula:
$(\sqrt{x} + \sqrt{7})(\sqrt{x} - \sqrt{7}) = (\sqrt{x})^2 - (\sqrt{7})^2 = x - 7$

STEP 6

Combine the simplified numerator and denominator:
$\frac{x - \sqrt{7x}}{x - 7}$
This is the rationalized form of the expression.
The rationalized form of the expression is:
$\boxed{\frac{x - \sqrt{7x}}{x - 7}}$
This corresponds to option a.

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QuestionWhich is the rationalized form of the expression xx+7\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7}}x​+7​x​​ ? a. x−7xx−7\frac{x-\sqrt{7 x}}{x-7}x−7x−7x​​ c. x+7xx−7\frac{x+\sqrt{7 x}}{x-7}x−7x+7x​​ b. 7x7\frac{\sqrt{7 x}}{7}77x​​ d. xx+7\frac{x}{x+7}x+7x​

Studdy Solution

STEP 1

1. We are given an expression with a radical in the denominator.2. The goal is to rationalize the denominator.3. Rationalizing involves eliminating the square root from the denominator.

STEP 2

1. Multiply the numerator and the denominator by the conjugate of the denominator.2. Simplify the resulting expression.

STEP 3

STEP 4

Simplify the numerator:x×(x−7)=x−7x\sqrt{x} \times (\sqrt{x} - \sqrt{7}) = x - \sqrt{7x}x​×(x​−7​)=x−7x​

STEP 5

Simplify the denominator using the difference of squares formula:(x+7)(x−7)=(x)2−(7)2=x−7(\sqrt{x} + \sqrt{7})(\sqrt{x} - \sqrt{7}) = (\sqrt{x})^2 - (\sqrt{7})^2 = x - 7(x​+7​)(x​−7​)=(x​)2−(7​)2=x−7

STEP 6

Combine the simplified numerator and denominator:x−7xx−7\frac{x - \sqrt{7x}}{x - 7}x−7x−7x​​This is the rationalized form of the expression. The rationalized form of the expression is:x−7xx−7 \boxed{\frac{x - \sqrt{7x}}{x - 7}} x−7x−7x​​​This corresponds to option a.

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QuestionWhich is the rationalized form of the expression $\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7}}$ ? a. $\frac{x-\sqrt{7 x}}{x-7}$ c. $\frac{x+\sqrt{7 x}}{x-7}$ b. $\frac{\sqrt{7 x}}{7}$ d. $\frac{x}{x+7}$

1. We are given an expression with a radical in the denominator.
2. The goal is to rationalize the denominator.
3. Rationalizing involves eliminating the square root from the denominator.

1. Multiply the numerator and the denominator by the conjugate of the denominator.
2. Simplify the resulting expression.

Simplify the numerator:
$\sqrt{x} \times (\sqrt{x} - \sqrt{7}) = x - \sqrt{7x}$

Simplify the denominator using the difference of squares formula:
$(\sqrt{x} + \sqrt{7})(\sqrt{x} - \sqrt{7}) = (\sqrt{x})^2 - (\sqrt{7})^2 = x - 7$

Combine the simplified numerator and denominator:
$\frac{x - \sqrt{7x}}{x - 7}$
This is the rationalized form of the expression.
The rationalized form of the expression is:
$\boxed{\frac{x - \sqrt{7x}}{x - 7}}$
This corresponds to option a.