Math / Algebra

QuestionEvaluate the expression $\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}}$ and write the result in the form $a+b i$ The real number $a$ equals $\square$ The real number $b$ equals $\square$

Studdy Solution

STEP 1

1. The expression $\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}}$ involves complex numbers, where $i$ is the imaginary unit with the property $i^2 = -1$ .
2. The square root of a negative number can be expressed in terms of $i$ , so $\sqrt{-x} = \sqrt{x} \cdot i$ for any positive real number $x$ .
3. The final expression should be in the form $a + bi$ , where $a$ and $b$ are real numbers.

STEP 2

1. Express the square roots of negative numbers in terms of $i$ .
2. Simplify the expression in the numerator and denominator separately.
3. Combine the results to simplify the overall expression.
4. Write the final result in the form $a + bi$ .

STEP 3

Express $\sqrt{-6}$ in terms of $i$ .
$\sqrt{-6} = \sqrt{6} \cdot i$

STEP 4

Express $\sqrt{-7}$ in terms of $i$ .
$\sqrt{-7} = \sqrt{7} \cdot i$

STEP 5

Simplify the denominator $\sqrt{-7} \cdot \sqrt{-7}$ .
$\sqrt{-7} \cdot \sqrt{-7} = (\sqrt{7} \cdot i) \cdot (\sqrt{7} \cdot i) = 7i^2 = 7(-1) = -7$

STEP 6

Combine the results from STEP_1 and STEP_3 to simplify the overall expression.
$\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}} = \frac{\sqrt{6} \cdot i}{-7} = -\frac{\sqrt{6}}{7} \cdot i$

STEP 7

Write the final result in the form $a + bi$ .
$a = 0$ $b = -\frac{\sqrt{6}}{7}$
Solution: The real number $a$ equals $0$ . The real number $b$ equals $-\frac{\sqrt{6}}{7}$ .

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QuestionEvaluate the expression $\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}}$ and write the result in the form $a+b i$ The real number $a$ equals $\square$ The real number $b$ equals $\square$

Studdy Solution

STEP 1

STEP 2

1. Express the square roots of negative numbers in terms of $i$ .
2. Simplify the expression in the numerator and denominator separately.
3. Combine the results to simplify the overall expression.
4. Write the final result in the form $a + bi$ .

STEP 3

Express $\sqrt{-6}$ in terms of $i$ .
$\sqrt{-6} = \sqrt{6} \cdot i$

STEP 4

Express $\sqrt{-7}$ in terms of $i$ .
$\sqrt{-7} = \sqrt{7} \cdot i$

STEP 5

Simplify the denominator $\sqrt{-7} \cdot \sqrt{-7}$ .
$\sqrt{-7} \cdot \sqrt{-7} = (\sqrt{7} \cdot i) \cdot (\sqrt{7} \cdot i) = 7i^2 = 7(-1) = -7$

STEP 6

Combine the results from STEP_1 and STEP_3 to simplify the overall expression.
$\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}} = \frac{\sqrt{6} \cdot i}{-7} = -\frac{\sqrt{6}}{7} \cdot i$

STEP 7

Write the final result in the form $a + bi$ .
$a = 0$ $b = -\frac{\sqrt{6}}{7}$
Solution: The real number $a$ equals $0$ . The real number $b$ equals $-\frac{\sqrt{6}}{7}$ .

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QuestionEvaluate the expression −6−7−7\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}}−7​−7​−6​​ and write the result in the form a+bia+b ia+bi The real number aaa equals □\square□ The real number bbb equals □\square□

Studdy Solution

STEP 1

STEP 2

1. Express the square roots of negative numbers in terms of iii.2. Simplify the expression in the numerator and denominator separately.3. Combine the results to simplify the overall expression.4. Write the final result in the form a+bia + bia+bi.

STEP 3

Express −6\sqrt{-6}−6​ in terms of iii.−6=6⋅i \sqrt{-6} = \sqrt{6} \cdot i −6​=6​⋅i

STEP 4

Express −7\sqrt{-7}−7​ in terms of iii.−7=7⋅i \sqrt{-7} = \sqrt{7} \cdot i −7​=7​⋅i

STEP 5

Simplify the denominator −7⋅−7\sqrt{-7} \cdot \sqrt{-7}−7​⋅−7​.−7⋅−7=(7⋅i)⋅(7⋅i)=7i2=7(−1)=−7 \sqrt{-7} \cdot \sqrt{-7} = (\sqrt{7} \cdot i) \cdot (\sqrt{7} \cdot i) = 7i^2 = 7(-1) = -7 −7​⋅−7​=(7​⋅i)⋅(7​⋅i)=7i2=7(−1)=−7

STEP 6

Combine the results from STEP_1 and STEP_3 to simplify the overall expression.−6−7−7=6⋅i−7=−67⋅i \frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}} = \frac{\sqrt{6} \cdot i}{-7} = -\frac{\sqrt{6}}{7} \cdot i −7​−7​−6​​=−76​⋅i​=−76​​⋅i

STEP 7

Write the final result in the form a+bia + bia+bi. a=0 a = 0 a=0 b=−67 b = -\frac{\sqrt{6}}{7} b=−76​​Solution: The real number aaa equals 000. The real number bbb equals −67-\frac{\sqrt{6}}{7}−76​​.

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QuestionEvaluate the expression $\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}}$ and write the result in the form $a+b i$ The real number $a$ equals $\square$ The real number $b$ equals $\square$

1. Express the square roots of negative numbers in terms of $i$ .
2. Simplify the expression in the numerator and denominator separately.
3. Combine the results to simplify the overall expression.
4. Write the final result in the form $a + bi$ .

Express $\sqrt{-6}$ in terms of $i$ .
$\sqrt{-6} = \sqrt{6} \cdot i$

Express $\sqrt{-7}$ in terms of $i$ .
$\sqrt{-7} = \sqrt{7} \cdot i$

Simplify the denominator $\sqrt{-7} \cdot \sqrt{-7}$ .
$\sqrt{-7} \cdot \sqrt{-7} = (\sqrt{7} \cdot i) \cdot (\sqrt{7} \cdot i) = 7i^2 = 7(-1) = -7$

Combine the results from STEP_1 and STEP_3 to simplify the overall expression.
$\frac{\sqrt{-6}}{\sqrt{-7} \sqrt{-7}} = \frac{\sqrt{6} \cdot i}{-7} = -\frac{\sqrt{6}}{7} \cdot i$

Write the final result in the form $a + bi$ .
$a = 0$ $b = -\frac{\sqrt{6}}{7}$
Solution: The real number $a$ equals $0$ . The real number $b$ equals $-\frac{\sqrt{6}}{7}$ .