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Sagot :
Sure, let's simplify each of the given expressions step-by-step.
### 1. Simplify [tex]\(\frac{12 m^2 n^3}{9 m^3 n}\)[/tex]
Let's break down the expression:
[tex]\[ \frac{12 m^2 n^3}{9 m^3 n} \][/tex]
First, factor out the coefficients:
[tex]\[ \frac{12}{9} = \frac{4}{3} \][/tex]
Next, simplify the powers of [tex]\(m\)[/tex]:
[tex]\[ \frac{m^2}{m^3} = \frac{1}{m^1} = \frac{1}{m} \][/tex]
Then, simplify the powers of [tex]\(n\)[/tex]:
[tex]\[ \frac{n^3}{n} = n^2 \][/tex]
Putting it all together:
[tex]\[ \frac{12 m^2 n^3}{9 m^3 n} = \frac{4}{3} \cdot \frac{n^2}{m} = \frac{4 n^2}{3 m} \][/tex]
### 2. Simplify [tex]\(\frac{4}{3 n^4}\)[/tex]
In this case, the expression is already in its simplest form. So,
[tex]\[ \frac{4}{3 n^4} \][/tex]
### 3. Simplify [tex]\(\frac{10 m n^2}{6 m^2 n^4}\)[/tex]
Let's simplify this expression step-by-step:
[tex]\[ \frac{10 m n^2}{6 m^2 n^4} \][/tex]
Simplify the coefficients:
[tex]\[ \frac{10}{6} = \frac{5}{3} \][/tex]
Then, simplify the powers of [tex]\(m\)[/tex]:
[tex]\[ \frac{m}{m^2} = \frac{1}{m^1} = \frac{1}{m} \][/tex]
Finally, simplify the powers of [tex]\(n\)[/tex]:
[tex]\[ \frac{n^2}{n^4} = \frac{1}{n^2} \][/tex]
Combining all parts:
[tex]\[ \frac{10 m n^2}{6 m^2 n^4} = \frac{5}{3} \cdot \frac{1}{m} \cdot \frac{1}{n^2} = \frac{5}{3 m n^2} \][/tex]
### 4. Simplify [tex]\(\frac{-3 m^6 n}{5 m^4 n^5}\)[/tex]
Simplify step-by-step:
[tex]\[ \frac{-3 m^6 n}{5 m^4 n^5} \][/tex]
Simplify the coefficients:
[tex]\[ \frac{-3}{5} = -\frac{3}{5} \][/tex]
Then, simplify the powers of [tex]\(m\)[/tex]:
[tex]\[ \frac{m^6}{m^4} = m^2 \][/tex]
Finally, simplify the powers of [tex]\(n\)[/tex]:
[tex]\[ \frac{n}{n^5} = \frac{1}{n^4} \][/tex]
Putting it all together:
[tex]\[ \frac{-3 m^6 n}{5 m^4 n^5} = -\frac{3}{5} \cdot m^2 \cdot \frac{1}{n^4} = -\frac{3 m^2}{5 n^4} \][/tex]
### 5. Simplify [tex]\(\frac{m^2 - 1}{m^2 + 5 m}\)[/tex]
Factorize the numerator and the denominator:
[tex]\[ m^2 - 1 = (m + 1)(m - 1) \][/tex]
[tex]\[ m^2 + 5m = m(m + 5) \][/tex]
Putting it all together:
[tex]\[ \frac{m^2 - 1}{m^2 + 5 m} = \frac{(m + 1)(m - 1)}{m(m + 5)} \][/tex]
Since these can't be simplified further, it remains:
[tex]\[ \frac{m^2 - 1}{m^2 + 5 m} = \frac{(m + 1)(m - 1)}{m(m + 5)} \][/tex]
### 6. Simplify [tex]\(\frac{m + 1}{m}\)[/tex]
This can be simplified straightforwardly:
[tex]\[ \frac{m + 1}{m} = \frac{m}{m} + \frac{1}{m} = 1 + \frac{1}{m} \][/tex]
So, it remains:
[tex]\[ \frac{m + 1}{m} \][/tex]
### 7. Simplify [tex]\(\frac{m^2 + 4 m - 5}{m^2 - 2 m + 1}\)[/tex]
Factorize the numerator and the denominator:
[tex]\[ m^2 + 4m - 5 = (m + 5)(m - 1) \][/tex]
[tex]\[ m^2 - 2m + 1 = (m - 1)^2 \][/tex]
Putting it all together:
[tex]\[ \frac{m^2 + 4 m - 5}{m^2 - 2 m + 1} = \frac{(m + 5)(m - 1)}{(m - 1)^2} \][/tex]
We can cancel [tex]\((m - 1)\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{(m + 5)(m - 1)}{(m - 1)(m - 1)} = \frac{m + 5}{m - 1} \][/tex]
### Summary
Combining all the simplified expressions:
[tex]\[ \frac{12 m^2 n^3}{9 m^3 n} = \frac{4 n^2}{3 m} \][/tex]
[tex]\[ \frac{4}{3 n^4} \][/tex]
[tex]\[ \frac{10 m n^2}{6 m^2 n^4} = \frac{5}{3 m n^2} \][/tex]
[tex]\[ \frac{-3 m^6 n}{5 m^4 n^5} = -\frac{3 m^2}{5 n^4} \][/tex]
[tex]\[ \frac{m^2 - 1}{m^2 + 5 m} = \frac{(m + 1)(m - 1)}{m(m + 5)} \][/tex]
[tex]\[ \frac{m + 1}{m} \][/tex]
[tex]\[ \frac{m^2 + 4 m - 5}{m^2 - 2 m + 1} = \frac{m + 5}{m - 1} \][/tex]
### 1. Simplify [tex]\(\frac{12 m^2 n^3}{9 m^3 n}\)[/tex]
Let's break down the expression:
[tex]\[ \frac{12 m^2 n^3}{9 m^3 n} \][/tex]
First, factor out the coefficients:
[tex]\[ \frac{12}{9} = \frac{4}{3} \][/tex]
Next, simplify the powers of [tex]\(m\)[/tex]:
[tex]\[ \frac{m^2}{m^3} = \frac{1}{m^1} = \frac{1}{m} \][/tex]
Then, simplify the powers of [tex]\(n\)[/tex]:
[tex]\[ \frac{n^3}{n} = n^2 \][/tex]
Putting it all together:
[tex]\[ \frac{12 m^2 n^3}{9 m^3 n} = \frac{4}{3} \cdot \frac{n^2}{m} = \frac{4 n^2}{3 m} \][/tex]
### 2. Simplify [tex]\(\frac{4}{3 n^4}\)[/tex]
In this case, the expression is already in its simplest form. So,
[tex]\[ \frac{4}{3 n^4} \][/tex]
### 3. Simplify [tex]\(\frac{10 m n^2}{6 m^2 n^4}\)[/tex]
Let's simplify this expression step-by-step:
[tex]\[ \frac{10 m n^2}{6 m^2 n^4} \][/tex]
Simplify the coefficients:
[tex]\[ \frac{10}{6} = \frac{5}{3} \][/tex]
Then, simplify the powers of [tex]\(m\)[/tex]:
[tex]\[ \frac{m}{m^2} = \frac{1}{m^1} = \frac{1}{m} \][/tex]
Finally, simplify the powers of [tex]\(n\)[/tex]:
[tex]\[ \frac{n^2}{n^4} = \frac{1}{n^2} \][/tex]
Combining all parts:
[tex]\[ \frac{10 m n^2}{6 m^2 n^4} = \frac{5}{3} \cdot \frac{1}{m} \cdot \frac{1}{n^2} = \frac{5}{3 m n^2} \][/tex]
### 4. Simplify [tex]\(\frac{-3 m^6 n}{5 m^4 n^5}\)[/tex]
Simplify step-by-step:
[tex]\[ \frac{-3 m^6 n}{5 m^4 n^5} \][/tex]
Simplify the coefficients:
[tex]\[ \frac{-3}{5} = -\frac{3}{5} \][/tex]
Then, simplify the powers of [tex]\(m\)[/tex]:
[tex]\[ \frac{m^6}{m^4} = m^2 \][/tex]
Finally, simplify the powers of [tex]\(n\)[/tex]:
[tex]\[ \frac{n}{n^5} = \frac{1}{n^4} \][/tex]
Putting it all together:
[tex]\[ \frac{-3 m^6 n}{5 m^4 n^5} = -\frac{3}{5} \cdot m^2 \cdot \frac{1}{n^4} = -\frac{3 m^2}{5 n^4} \][/tex]
### 5. Simplify [tex]\(\frac{m^2 - 1}{m^2 + 5 m}\)[/tex]
Factorize the numerator and the denominator:
[tex]\[ m^2 - 1 = (m + 1)(m - 1) \][/tex]
[tex]\[ m^2 + 5m = m(m + 5) \][/tex]
Putting it all together:
[tex]\[ \frac{m^2 - 1}{m^2 + 5 m} = \frac{(m + 1)(m - 1)}{m(m + 5)} \][/tex]
Since these can't be simplified further, it remains:
[tex]\[ \frac{m^2 - 1}{m^2 + 5 m} = \frac{(m + 1)(m - 1)}{m(m + 5)} \][/tex]
### 6. Simplify [tex]\(\frac{m + 1}{m}\)[/tex]
This can be simplified straightforwardly:
[tex]\[ \frac{m + 1}{m} = \frac{m}{m} + \frac{1}{m} = 1 + \frac{1}{m} \][/tex]
So, it remains:
[tex]\[ \frac{m + 1}{m} \][/tex]
### 7. Simplify [tex]\(\frac{m^2 + 4 m - 5}{m^2 - 2 m + 1}\)[/tex]
Factorize the numerator and the denominator:
[tex]\[ m^2 + 4m - 5 = (m + 5)(m - 1) \][/tex]
[tex]\[ m^2 - 2m + 1 = (m - 1)^2 \][/tex]
Putting it all together:
[tex]\[ \frac{m^2 + 4 m - 5}{m^2 - 2 m + 1} = \frac{(m + 5)(m - 1)}{(m - 1)^2} \][/tex]
We can cancel [tex]\((m - 1)\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{(m + 5)(m - 1)}{(m - 1)(m - 1)} = \frac{m + 5}{m - 1} \][/tex]
### Summary
Combining all the simplified expressions:
[tex]\[ \frac{12 m^2 n^3}{9 m^3 n} = \frac{4 n^2}{3 m} \][/tex]
[tex]\[ \frac{4}{3 n^4} \][/tex]
[tex]\[ \frac{10 m n^2}{6 m^2 n^4} = \frac{5}{3 m n^2} \][/tex]
[tex]\[ \frac{-3 m^6 n}{5 m^4 n^5} = -\frac{3 m^2}{5 n^4} \][/tex]
[tex]\[ \frac{m^2 - 1}{m^2 + 5 m} = \frac{(m + 1)(m - 1)}{m(m + 5)} \][/tex]
[tex]\[ \frac{m + 1}{m} \][/tex]
[tex]\[ \frac{m^2 + 4 m - 5}{m^2 - 2 m + 1} = \frac{m + 5}{m - 1} \][/tex]
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