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QuestionSimplify the function: m2+6m+s2m12+32m\frac{\frac{m}{2}+\frac{6 m+s}{2 m}}{\frac{1}{2}+\frac{3}{2 m}}

Studdy Solution

STEP 1

Assumptions1. The variable m is a non-zero real number. The variable s is a real number

STEP 2

First, we need to simplify the numerator and the denominator separately. Let's start with the numerator.The numerator ism2+6m+s2m\frac{m}{2}+\frac{6 m+s}{2 m}

STEP 3

To simplify the numerator, we need to find a common denominator for the two fractions. The common denominator is2m.
So, rewrite the first fraction with2m as the denominatorm×m2×m+6m+s2m\frac{m \times m}{2 \times m}+\frac{6 m+s}{2 m}

STEP 4

implify the first fractionm22m+6m+s2m\frac{m^2}{2m}+\frac{6 m+s}{2 m}

STEP 5

Now, we can add the two fractions togetherm2+m+s2m\frac{m^2 +m + s}{2m}

STEP 6

Next, let's simplify the denominator. The denominator is12+32m\frac{1}{2}+\frac{3}{2 m}

STEP 7

To simplify the denominator, we need to find a common denominator for the two fractions. The common denominator is2m.
So, rewrite the first fraction with2m as the denominator1×m2×m+32m\frac{1 \times m}{2 \times m}+\frac{3}{2 m}

STEP 8

implify the first fractionm2m+32m\frac{m}{2m}+\frac{3}{2 m}

STEP 9

Now, we can add the two fractions togetherm+32m\frac{m +3}{2m}

STEP 10

Now that we have simplified the numerator and the denominator, we can rewrite the original complex fractionm2+6m+s2mm+32m\frac{\frac{m^2 +6m + s}{2m}}{\frac{m +3}{2m}}

STEP 11

To simplify a complex fraction, we can multiply the numerator and the denominator by the reciprocal of the denominator. The reciprocal of the denominator ismm+3\frac{m}{m +3}So, multiply the numerator and the denominator by this reciprocalm+6m+sm×mm+3m+3m×mm+3\frac{\frac{m^ +6m + s}{m} \times \frac{m}{m +3}}{\frac{m +3}{m} \times \frac{m}{m +3}}

STEP 12

implify the multiplicationm2+6m+sm+\frac{m^2 +6m + s}{m +}This is the simplified form of the original complex fraction.

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