QuestionSimplify the imaginary number $7 \sqrt{-\frac{7}{125}}$ .

Studdy Solution

STEP 1

Assumptions1. We are given an imaginary number $7 \sqrt{-\frac{7}{125}}$ . We know that the square root of a negative number is an imaginary number. Specifically, $\sqrt{-1} = i$ .

STEP 2

We first separate the negative sign from the fraction under the square root. The negative sign under the square root becomes $i$ .
$7 \sqrt{-\frac{7}{125}} =7i \sqrt{\frac{7}{125}}$

STEP 3

Next, we simplify the square root by finding the square root of the numerator and the denominator separately.
$7i \sqrt{\frac{7}{125}} =7i \frac{\sqrt{7}}{\sqrt{125}}$

STEP 4

We know that $\sqrt{125} =\sqrt{}$ , so we substitute this into the equation.
$7i \frac{\sqrt{7}}{\sqrt{125}} =7i \frac{\sqrt{7}}{\sqrt{}}$

STEP 5

We simplify the equation by cancelling out the common factors.
$7i \frac{\sqrt{7}}{5\sqrt{5}} = \frac{7i\sqrt{7}}{5\sqrt{5}}$

STEP 6

We can simplify further by rationalizing the denominator. We do this by multiplying the numerator and the denominator by $\sqrt{5}$ .
$\frac{i\sqrt{}}{5\sqrt{5}} = \frac{i\sqrt{}\sqrt{5}}{5\sqrt{5}\sqrt{5}}$

STEP 7

implify the equation by cancelling out the common factors.
$\frac{7i\sqrt{7}\sqrt{5}}{5\sqrt{5}\sqrt{5}} = \frac{7i\sqrt{35}}{25}$ So, the simplified form of the given imaginary number is $\frac{7i\sqrt{35}}{25}$ .

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