QuestionSimplify: $\frac{5}{2-i}$ . Choose the correct answer: a. $2+i$ , b. $\frac{10+5 i}{3}$ , c. $2-i$ , d. $\frac{10-5 i}{3}$ .

Studdy Solution

STEP 1

Assumptions1. We are given a complex number in the denominator of the fraction. . We need to simplify the fraction completely.

STEP 2

To simplify a fraction with a complex number in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$ .

STEP 3

The conjugate of $2 - i$ is $2 + i$ . Multiply both the numerator and the denominator by this conjugate.
$\frac{5}{2-i} \times \frac{2+i}{2+i}$

STEP 4

Multiply the numerators together and the denominators together.
$\frac{ \times (2+i)}{(2-i) \times (2+i)}$

STEP 5

istribute the5 in the numerator and use the formula $(a-b)(a+b) = a^2 - b^2$ in the denominator.
$\frac{10+5i}{2^2 - (-i)^2}$

STEP 6

implify the denominator. Remember that $i^2 = -1$ .
$\frac{10+5i}{4 - (-1)}$

STEP 7

implify the denominator further.
$\frac{10+5i}{5}$

STEP 8

Divide each term in the numerator by the denominator.
$\frac{10}{5} + \frac{5i}{5}$

STEP 9

implify each term.
$2+i$ So, the simplified form of $\frac{5}{2-i}$ is $2+i$ .

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QuestionSimplify: $\frac{5}{2-i}$ . Choose the correct answer: a. $2+i$ , b. $\frac{10+5 i}{3}$ , c. $2-i$ , d. $\frac{10-5 i}{3}$ .

Studdy Solution

STEP 1

Assumptions1. We are given a complex number in the denominator of the fraction. . We need to simplify the fraction completely.

STEP 2

To simplify a fraction with a complex number in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$ .

STEP 3

The conjugate of $2 - i$ is $2 + i$ . Multiply both the numerator and the denominator by this conjugate.
$\frac{5}{2-i} \times \frac{2+i}{2+i}$

STEP 4

Multiply the numerators together and the denominators together.
$\frac{ \times (2+i)}{(2-i) \times (2+i)}$

STEP 5

istribute the5 in the numerator and use the formula $(a-b)(a+b) = a^2 - b^2$ in the denominator.
$\frac{10+5i}{2^2 - (-i)^2}$

STEP 6

implify the denominator. Remember that $i^2 = -1$ .
$\frac{10+5i}{4 - (-1)}$

STEP 7

implify the denominator further.
$\frac{10+5i}{5}$

STEP 8

Divide each term in the numerator by the denominator.
$\frac{10}{5} + \frac{5i}{5}$

STEP 9

implify each term.
$2+i$ So, the simplified form of $\frac{5}{2-i}$ is $2+i$ .

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QuestionSimplify: 52−i\frac{5}{2-i}2−i5​. Choose the correct answer: a. 2+i2+i2+i, b. 10+5i3\frac{10+5 i}{3}310+5i​, c. 2−i2-i2−i, d. 10−5i3\frac{10-5 i}{3}310−5i​.

Studdy Solution

STEP 1

Assumptions1. We are given a complex number in the denominator of the fraction. . We need to simplify the fraction completely.

STEP 2

To simplify a fraction with a complex number in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a+bia + bia+bi is a−bia - bia−bi.

STEP 3

The conjugate of 2−i2 - i2−i is 2+i2 + i2+i. Multiply both the numerator and the denominator by this conjugate.52−i×2+i2+i\frac{5}{2-i} \times \frac{2+i}{2+i}2−i5​×2+i2+i​

STEP 4

Multiply the numerators together and the denominators together.×(2+i)(2−i)×(2+i)\frac{ \times (2+i)}{(2-i) \times (2+i)}(2−i)×(2+i)×(2+i)​

STEP 5

istribute the5 in the numerator and use the formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2(a−b)(a+b)=a2−b2 in the denominator.10+5i22−(−i)2\frac{10+5i}{2^2 - (-i)^2}22−(−i)210+5i​

STEP 6

implify the denominator. Remember that i2=−1i^2 = -1i2=−1.10+5i4−(−1)\frac{10+5i}{4 - (-1)}4−(−1)10+5i​

STEP 7

implify the denominator further.10+5i5\frac{10+5i}{5}510+5i​

STEP 8

Divide each term in the numerator by the denominator.105+5i5\frac{10}{5} + \frac{5i}{5}510​+55i​

STEP 9

implify each term.2+i2+i2+iSo, the simplified form of 52−i\frac{5}{2-i}2−i5​ is 2+i2+i2+i.

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QuestionSimplify: $\frac{5}{2-i}$ . Choose the correct answer: a. $2+i$ , b. $\frac{10+5 i}{3}$ , c. $2-i$ , d. $\frac{10-5 i}{3}$ .

To simplify a fraction with a complex number in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$ .

The conjugate of $2 - i$ is $2 + i$ . Multiply both the numerator and the denominator by this conjugate.
$\frac{5}{2-i} \times \frac{2+i}{2+i}$

Multiply the numerators together and the denominators together.
$\frac{ \times (2+i)}{(2-i) \times (2+i)}$

istribute the5 in the numerator and use the formula $(a-b)(a+b) = a^2 - b^2$ in the denominator.
$\frac{10+5i}{2^2 - (-i)^2}$

implify the denominator. Remember that $i^2 = -1$ .
$\frac{10+5i}{4 - (-1)}$

implify the denominator further.
$\frac{10+5i}{5}$

Divide each term in the numerator by the denominator.
$\frac{10}{5} + \frac{5i}{5}$

implify each term.
$2+i$ So, the simplified form of $\frac{5}{2-i}$ is $2+i$ .