$\frac{1+\sqrt{2}}{\sqrt{3}+2}$
Answer (2)
We simplified the expression \frac{1+\text{\sqrt{2}}}{\text{\sqrt{3}}+2} by rationalizing the denominator, leading to the final form of 2 − 3 − 6 + 2 2 . The approximate value of this expression is 0.647 .
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Rationalize the denominator of the expression by multiplying the numerator and denominator by the conjugate of the denominator.
Simplify the denominator: ( 3 + 2 ) ( 3 − 2 ) = − 1 .
Expand the numerator: ( 1 + 2 ) ( 3 − 2 ) = 3 − 2 + 6 − 2 2 .
Simplify the expression to 2 − 3 − 6 + 2 2 , which is approximately 0.647 .
The final simplified expression is 2 − 3 − 6 + 2 2 ≈ 0.647 .
Explanation
Understanding the Expression We are given the expression 3 + 2 1 + 2 . Our goal is to simplify this expression and find its approximate value.
Rationalizing the Denominator To simplify the expression, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is 3 − 2 . This gives us: 3 + 2 1 + 2 = ( 3 + 2 ) ( 3 − 2 ) ( 1 + 2 ) ( 3 − 2 )
Simplifying the Denominator Now, we simplify the denominator: ( 3 + 2 ) ( 3 − 2 ) = ( 3 ) 2 − 2 2 = 3 − 4 = − 1
Expanding the Numerator Next, we expand the numerator: ( 1 + 2 ) ( 3 − 2 ) = 1 ( 3 ) + 1 ( − 2 ) + 2 ( 3 ) + 2 ( − 2 ) = 3 − 2 + 6 − 2 2
Simplifying the Expression So, the expression becomes: − 1 3 − 2 + 6 − 2 2 = 2 − 3 − 6 + 2 2
Approximating the Value Finally, we can approximate the value of the expression: 2 − 3 − 6 + 2 2 ≈ 2 − 1.732 − 2.449 + 2 ( 1.414 ) ≈ 2 − 1.732 − 2.449 + 2.828 ≈ 0.647 Thus, the simplified expression is 2 − 3 − 6 + 2 2 , and its approximate value is 0.647 .
Examples
Rationalizing denominators is a useful technique in various fields, such as electrical engineering when dealing with impedance calculations or in physics when working with wave functions. For example, if you are calculating the total impedance of a circuit and end up with an expression like 2 + 1 1 , you would rationalize the denominator to simplify further calculations. This makes it easier to combine with other impedance values and analyze the circuit's behavior. The simplified form, 2 − 1 , allows for easier numerical computations and a clearer understanding of the impedance value.
