Math / Algebra

QuestionRationalize the denominator and simplify: $\frac{\sqrt{5}}{a-\sqrt{5}}$ $-\frac{1}{a}$ $\frac{\sqrt{5} a+\sqrt{5}}{a^{2}-5}$ $a+\sqrt{5}$ $\frac{\sqrt{5} a+5}{a^{2}-5}$

Studdy Solution

STEP 1

1. We need to rationalize the denominator of the expression $\frac{\sqrt{5}}{a-\sqrt{5}}$ .
2. Rationalizing involves eliminating the square root from the denominator.
3. We will use the conjugate of the denominator to rationalize.

STEP 2

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

STEP 3

Identify the conjugate of the denominator $a - \sqrt{5}$ . The conjugate is $a + \sqrt{5}$ .

STEP 4

Multiply both the numerator and the denominator by the conjugate $a + \sqrt{5}$ :
$\frac{\sqrt{5}}{a-\sqrt{5}} \times \frac{a+\sqrt{5}}{a+\sqrt{5}} = \frac{\sqrt{5}(a+\sqrt{5})}{(a-\sqrt{5})(a+\sqrt{5})}$

STEP 5

Simplify the denominator using the difference of squares formula:
$(a-\sqrt{5})(a+\sqrt{5}) = a^2 - (\sqrt{5})^2 = a^2 - 5$

STEP 6

Simplify the numerator:
$\sqrt{5}(a+\sqrt{5}) = \sqrt{5}a + 5$

STEP 7

Combine the simplified numerator and denominator:
$\frac{\sqrt{5}a + 5}{a^2 - 5}$
The rationalized and simplified expression is:
$\frac{\sqrt{5}a + 5}{a^2 - 5}$

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QuestionRationalize the denominator and simplify: $\frac{\sqrt{5}}{a-\sqrt{5}}$ $-\frac{1}{a}$ $\frac{\sqrt{5} a+\sqrt{5}}{a^{2}-5}$ $a+\sqrt{5}$ $\frac{\sqrt{5} a+5}{a^{2}-5}$

Studdy Solution

STEP 1

1. We need to rationalize the denominator of the expression $\frac{\sqrt{5}}{a-\sqrt{5}}$ .
2. Rationalizing involves eliminating the square root from the denominator.
3. We will use the conjugate of the denominator to rationalize.

STEP 2

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

STEP 3

Identify the conjugate of the denominator $a - \sqrt{5}$ . The conjugate is $a + \sqrt{5}$ .

STEP 4

Multiply both the numerator and the denominator by the conjugate $a + \sqrt{5}$ :
$\frac{\sqrt{5}}{a-\sqrt{5}} \times \frac{a+\sqrt{5}}{a+\sqrt{5}} = \frac{\sqrt{5}(a+\sqrt{5})}{(a-\sqrt{5})(a+\sqrt{5})}$

STEP 5

Simplify the denominator using the difference of squares formula:
$(a-\sqrt{5})(a+\sqrt{5}) = a^2 - (\sqrt{5})^2 = a^2 - 5$

STEP 6

Simplify the numerator:
$\sqrt{5}(a+\sqrt{5}) = \sqrt{5}a + 5$

STEP 7

Combine the simplified numerator and denominator:
$\frac{\sqrt{5}a + 5}{a^2 - 5}$
The rationalized and simplified expression is:
$\frac{\sqrt{5}a + 5}{a^2 - 5}$

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QuestionRationalize the denominator and simplify: 5a−5\frac{\sqrt{5}}{a-\sqrt{5}}a−5​5​​ −1a-\frac{1}{a}−a1​ 5a+5a2−5\frac{\sqrt{5} a+\sqrt{5}}{a^{2}-5}a2−55​a+5​​ a+5a+\sqrt{5}a+5​ 5a+5a2−5\frac{\sqrt{5} a+5}{a^{2}-5}a2−55​a+5​

Studdy Solution

STEP 1

1. We need to rationalize the denominator of the expression 5a−5\frac{\sqrt{5}}{a-\sqrt{5}}a−5​5​​.2. Rationalizing involves eliminating the square root from the denominator.3. We will use the conjugate of the denominator to rationalize.

STEP 2

1. Identify the conjugate of the denominator.2. Multiply the numerator and the denominator by the conjugate.3. Simplify the resulting expression.

STEP 3

Identify the conjugate of the denominator a−5a - \sqrt{5}a−5​. The conjugate is a+5a + \sqrt{5}a+5​.

STEP 4

STEP 5

Simplify the denominator using the difference of squares formula:(a−5)(a+5)=a2−(5)2=a2−5(a-\sqrt{5})(a+\sqrt{5}) = a^2 - (\sqrt{5})^2 = a^2 - 5(a−5​)(a+5​)=a2−(5​)2=a2−5

STEP 6

Simplify the numerator:5(a+5)=5a+5\sqrt{5}(a+\sqrt{5}) = \sqrt{5}a + 55​(a+5​)=5​a+5

STEP 7

Combine the simplified numerator and denominator:5a+5a2−5\frac{\sqrt{5}a + 5}{a^2 - 5}a2−55​a+5​ The rationalized and simplified expression is:5a+5a2−5\frac{\sqrt{5}a + 5}{a^2 - 5}a2−55​a+5​

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QuestionRationalize the denominator and simplify: $\frac{\sqrt{5}}{a-\sqrt{5}}$ $-\frac{1}{a}$ $\frac{\sqrt{5} a+\sqrt{5}}{a^{2}-5}$ $a+\sqrt{5}$ $\frac{\sqrt{5} a+5}{a^{2}-5}$

1. We need to rationalize the denominator of the expression $\frac{\sqrt{5}}{a-\sqrt{5}}$ .
2. Rationalizing involves eliminating the square root from the denominator.
3. We will use the conjugate of the denominator to rationalize.

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

Identify the conjugate of the denominator $a - \sqrt{5}$ . The conjugate is $a + \sqrt{5}$ .

Simplify the denominator using the difference of squares formula:
$(a-\sqrt{5})(a+\sqrt{5}) = a^2 - (\sqrt{5})^2 = a^2 - 5$

Simplify the numerator:
$\sqrt{5}(a+\sqrt{5}) = \sqrt{5}a + 5$

Combine the simplified numerator and denominator:
$\frac{\sqrt{5}a + 5}{a^2 - 5}$
The rationalized and simplified expression is:
$\frac{\sqrt{5}a + 5}{a^2 - 5}$