QuestionSimplify $\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m/n + 1\right)$ .

Studdy Solution

STEP 1

Assumptions1. m and n are real numbers. n is not equal to0, as it is in the denominator

STEP 2

First, we simplify the term $(-m)^{0}$ .
Any non-zero number raised to the power of0 is1.
$(-m)^{0} =1$

STEP 3

Now, we substitute $(-m)^{0}$ with1 in the original expression.
$\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m n^{-1}+1\right)$

STEP 4

Next, we simplify the term $\left(\frac{m}{n}\right)^{2}$ .
This is equivalent to $\frac{m^2}{n^2}$ .
$\left[\frac{m^2}{n^2}-1\right] \div\left(m n^{-1}+1\right)$

STEP 5

We simplify the term $m n^{-1}$ .
This is equivalent to $\frac{m}{n}$ .
$\left[\frac{m^2}{n^2}-1\right] \div\left(\frac{m}{n}+1\right)$

STEP 6

Now, we simplify the expression by subtracting1 from $\frac{m^2}{n^2}$ .
This is equivalent to $\frac{m^2-n^2}{n^2}$ .
$\frac{m^2-n^2}{n^2} \div\left(\frac{m}{n}+1\right)$

STEP 7

Next, we rewrite the division as a multiplication by the reciprocal.
$\frac{m^2-n^2}{n^2} \times\left(\frac{n}{m}+1\right)^{-1}$

STEP 8

We simplify the term $\left(\frac{n}{m}+1\right)^{-1}$ .
This is equivalent to $\frac{m}{m+n}$ .
$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n}$

STEP 9

Finally, we multiply the two fractions together.
$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n} = \frac{m(m^2-n^2)}{n^2(m+n)}$ The simplified expression is $\frac{m(m^2-n^2)}{n^2(m+n)}$ .

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QuestionSimplify $\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m/n + 1\right)$ .

Studdy Solution

STEP 1

Assumptions1. m and n are real numbers. n is not equal to0, as it is in the denominator

STEP 2

First, we simplify the term $(-m)^{0}$ .
Any non-zero number raised to the power of0 is1.
$(-m)^{0} =1$

STEP 3

Now, we substitute $(-m)^{0}$ with1 in the original expression.
$\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m n^{-1}+1\right)$

STEP 4

Next, we simplify the term $\left(\frac{m}{n}\right)^{2}$ .
This is equivalent to $\frac{m^2}{n^2}$ .
$\left[\frac{m^2}{n^2}-1\right] \div\left(m n^{-1}+1\right)$

STEP 5

We simplify the term $m n^{-1}$ .
This is equivalent to $\frac{m}{n}$ .
$\left[\frac{m^2}{n^2}-1\right] \div\left(\frac{m}{n}+1\right)$

STEP 6

Now, we simplify the expression by subtracting1 from $\frac{m^2}{n^2}$ .
This is equivalent to $\frac{m^2-n^2}{n^2}$ .
$\frac{m^2-n^2}{n^2} \div\left(\frac{m}{n}+1\right)$

STEP 7

Next, we rewrite the division as a multiplication by the reciprocal.
$\frac{m^2-n^2}{n^2} \times\left(\frac{n}{m}+1\right)^{-1}$

STEP 8

We simplify the term $\left(\frac{n}{m}+1\right)^{-1}$ .
This is equivalent to $\frac{m}{m+n}$ .
$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n}$

STEP 9

Finally, we multiply the two fractions together.
$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n} = \frac{m(m^2-n^2)}{n^2(m+n)}$ The simplified expression is $\frac{m(m^2-n^2)}{n^2(m+n)}$ .

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QuestionSimplify [(mn)2−1]÷(m/n+1)\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m/n + 1\right)[(nm​)2−1]÷(m/n+1).

Studdy Solution

STEP 1

Assumptions1. m and n are real numbers. n is not equal to0, as it is in the denominator

STEP 2

First, we simplify the term (−m)0(-m)^{0}(−m)0.Any non-zero number raised to the power of0 is1.(−m)0=1(-m)^{0} =1(−m)0=1

STEP 3

Now, we substitute (−m)0(-m)^{0}(−m)0 with1 in the original expression.[(mn)2−1]÷(mn−1+1)\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m n^{-1}+1\right)[(nm​)2−1]÷(mn−1+1)

STEP 4

Next, we simplify the term (mn)2\left(\frac{m}{n}\right)^{2}(nm​)2.This is equivalent to m2n2\frac{m^2}{n^2}n2m2​.[m2n2−1]÷(mn−1+1)\left[\frac{m^2}{n^2}-1\right] \div\left(m n^{-1}+1\right)[n2m2​−1]÷(mn−1+1)

STEP 5

We simplify the term mn−1m n^{-1}mn−1.This is equivalent to mn\frac{m}{n}nm​.[m2n2−1]÷(mn+1)\left[\frac{m^2}{n^2}-1\right] \div\left(\frac{m}{n}+1\right)[n2m2​−1]÷(nm​+1)

STEP 6

Now, we simplify the expression by subtracting1 from m2n2\frac{m^2}{n^2}n2m2​.This is equivalent to m2−n2n2\frac{m^2-n^2}{n^2}n2m2−n2​.m2−n2n2÷(mn+1)\frac{m^2-n^2}{n^2} \div\left(\frac{m}{n}+1\right)n2m2−n2​÷(nm​+1)

STEP 7

Next, we rewrite the division as a multiplication by the reciprocal.m2−n2n2×(nm+1)−1\frac{m^2-n^2}{n^2} \times\left(\frac{n}{m}+1\right)^{-1}n2m2−n2​×(mn​+1)−1

STEP 8

We simplify the term (nm+1)−1\left(\frac{n}{m}+1\right)^{-1}(mn​+1)−1.This is equivalent to mm+n\frac{m}{m+n}m+nm​.m2−n2n2×mm+n\frac{m^2-n^2}{n^2} \times\frac{m}{m+n}n2m2−n2​×m+nm​

STEP 9

Finally, we multiply the two fractions together.m2−n2n2×mm+n=m(m2−n2)n2(m+n)\frac{m^2-n^2}{n^2} \times\frac{m}{m+n} = \frac{m(m^2-n^2)}{n^2(m+n)}n2m2−n2​×m+nm​=n2(m+n)m(m2−n2)​The simplified expression is m(m2−n2)n2(m+n)\frac{m(m^2-n^2)}{n^2(m+n)}n2(m+n)m(m2−n2)​.

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QuestionSimplify $\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m/n + 1\right)$ .

First, we simplify the term $(-m)^{0}$ .
Any non-zero number raised to the power of0 is1.
$(-m)^{0} =1$

Now, we substitute $(-m)^{0}$ with1 in the original expression.
$\left[\left(\frac{m}{n}\right)^{2}-1\right] \div\left(m n^{-1}+1\right)$

Next, we simplify the term $\left(\frac{m}{n}\right)^{2}$ .
This is equivalent to $\frac{m^2}{n^2}$ .
$\left[\frac{m^2}{n^2}-1\right] \div\left(m n^{-1}+1\right)$

We simplify the term $m n^{-1}$ .
This is equivalent to $\frac{m}{n}$ .
$\left[\frac{m^2}{n^2}-1\right] \div\left(\frac{m}{n}+1\right)$

Now, we simplify the expression by subtracting1 from $\frac{m^2}{n^2}$ .
This is equivalent to $\frac{m^2-n^2}{n^2}$ .
$\frac{m^2-n^2}{n^2} \div\left(\frac{m}{n}+1\right)$

Next, we rewrite the division as a multiplication by the reciprocal.
$\frac{m^2-n^2}{n^2} \times\left(\frac{n}{m}+1\right)^{-1}$

We simplify the term $\left(\frac{n}{m}+1\right)^{-1}$ .
This is equivalent to $\frac{m}{m+n}$ .
$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n}$

Finally, we multiply the two fractions together.
$\frac{m^2-n^2}{n^2} \times\frac{m}{m+n} = \frac{m(m^2-n^2)}{n^2(m+n)}$ The simplified expression is $\frac{m(m^2-n^2)}{n^2(m+n)}$ .