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Math  /  Algebra

QuestionExpress as a single fraction in simplest radical form with a rational denominator. 7+312+7\frac{\sqrt{7}+\sqrt{3}}{\sqrt{12}+\sqrt{7}}

Studdy Solution

STEP 1

1. We need to express the given expression as a single fraction.
2. The fraction should be in simplest radical form.
3. The denominator should be rationalized.

STEP 2

1. Simplify the square roots in the expression.
2. Rationalize the denominator.
3. Simplify the resulting expression.

STEP 3

Simplify the square roots in the expression. Notice that 12\sqrt{12} can be simplified:
12=4×3=4×3=23\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
So the expression becomes:
7+323+7\frac{\sqrt{7} + \sqrt{3}}{2\sqrt{3} + \sqrt{7}}

STEP 4

Rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, 23−72\sqrt{3} - \sqrt{7}:
(7+3)(23−7)(23+7)(23−7)\frac{(\sqrt{7} + \sqrt{3})(2\sqrt{3} - \sqrt{7})}{(2\sqrt{3} + \sqrt{7})(2\sqrt{3} - \sqrt{7})}

STEP 5

Calculate the denominator using the difference of squares formula:
(23)2−(7)2=4×3−7=12−7=5(2\sqrt{3})^2 - (\sqrt{7})^2 = 4 \times 3 - 7 = 12 - 7 = 5

STEP 6

Expand the numerator:
(7+3)(23−7)=7⋅23−7⋅7+3⋅23−3⋅7(\sqrt{7} + \sqrt{3})(2\sqrt{3} - \sqrt{7}) = \sqrt{7} \cdot 2\sqrt{3} - \sqrt{7} \cdot \sqrt{7} + \sqrt{3} \cdot 2\sqrt{3} - \sqrt{3} \cdot \sqrt{7}
=221−7+2⋅3−21= 2\sqrt{21} - 7 + 2\cdot3 - \sqrt{21}
=221−7+6−21= 2\sqrt{21} - 7 + 6 - \sqrt{21}
=21−1= \sqrt{21} - 1

STEP 7

Combine the results to form the simplified expression:
21−15\frac{\sqrt{21} - 1}{5}
This is the expression as a single fraction in simplest radical form with a rational denominator.

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