Math  /  Algebra

QuestionSimplify completely: 2+10m125m2\frac{2+\frac{10}{m}}{1-\frac{25}{m^{2}}}
Enter the numerator and denominator separately in the boxes below. If the denominator is 1 , enter the number 1. Do not leave either box blank. Make sure that the coefficient on the variable is positive. Answer: \square \square Numerator preview:

Studdy Solution

STEP 1

1. We are given a complex fraction and need to simplify it completely.
2. The expression involves a variable m m .
3. The goal is to express the fraction in its simplest form.

STEP 2

1. Simplify the numerator of the complex fraction.
2. Simplify the denominator of the complex fraction.
3. Combine and simplify the overall expression.

STEP 3

Simplify the numerator 2+10m 2 + \frac{10}{m} .
First, find a common denominator for the terms in the numerator. The common denominator is m m .
2=2mm 2 = \frac{2m}{m}
So, the numerator becomes:
2mm+10m=2m+10m \frac{2m}{m} + \frac{10}{m} = \frac{2m + 10}{m}

STEP 4

Simplify the denominator 125m2 1 - \frac{25}{m^2} .
Find a common denominator for the terms in the denominator. The common denominator is m2 m^2 .
1=m2m2 1 = \frac{m^2}{m^2}
So, the denominator becomes:
m2m225m2=m225m2 \frac{m^2}{m^2} - \frac{25}{m^2} = \frac{m^2 - 25}{m^2}

STEP 5

Combine the simplified numerator and denominator into a single fraction:
2m+10mm225m2 \frac{\frac{2m + 10}{m}}{\frac{m^2 - 25}{m^2}}
This can be simplified by multiplying by the reciprocal of the denominator:
=2m+10m×m2m225 = \frac{2m + 10}{m} \times \frac{m^2}{m^2 - 25}

STEP 6

Simplify the expression further by canceling common factors.
Notice that m m in the numerator and denominator can be canceled:
=(2m+10)mm(m225) = \frac{(2m + 10) \cdot m}{m \cdot (m^2 - 25)}
Cancel m m :
=2m+10m225 = \frac{2m + 10}{m^2 - 25}

STEP 7

Factor the expression m225 m^2 - 25 as it is a difference of squares:
m225=(m5)(m+5) m^2 - 25 = (m - 5)(m + 5)
So the expression becomes:
2m+10(m5)(m+5) \frac{2m + 10}{(m - 5)(m + 5)}
Factor out the common factor in the numerator:
2m+10=2(m+5) 2m + 10 = 2(m + 5)
Thus, the expression simplifies to:
2(m+5)(m5)(m+5) \frac{2(m + 5)}{(m - 5)(m + 5)}
Cancel the common factor (m+5) (m + 5) :
=2m5 = \frac{2}{m - 5}
The simplified numerator is: 2 2
The simplified denominator is: m5 m - 5

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