Basic Math Examples

Simplify ((n^2-m^2)/(2m-3n))÷((m-n)/(4m^2-9n^2))
n2-m22m-3n÷m-n4m2-9n2n2m22m3n÷mn4m29n2
Step 1
To divide by a fraction, multiply by its reciprocal.
n2-m22m-3n4m2-9n2m-nn2m22m3n4m29n2mn
Step 2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b)a2b2=(a+b)(ab) where a=na=n and b=mb=m.
(n+m)(n-m)2m-3n4m2-9n2m-n(n+m)(nm)2m3n4m29n2mn
Step 3
Simplify the numerator.
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Step 3.1
Rewrite 4m24m2 as (2m)2(2m)2.
(n+m)(n-m)2m-3n(2m)2-9n2m-n(n+m)(nm)2m3n(2m)29n2mn
Step 3.2
Rewrite 9n29n2 as (3n)2(3n)2.
(n+m)(n-m)2m-3n(2m)2-(3n)2m-n(n+m)(nm)2m3n(2m)2(3n)2mn
Step 3.3
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b)a2b2=(a+b)(ab) where a=2ma=2m and b=3nb=3n.
(n+m)(n-m)2m-3n(2m+3n)(2m-(3n))m-n(n+m)(nm)2m3n(2m+3n)(2m(3n))mn
Step 3.4
Multiply 33 by -11.
(n+m)(n-m)2m-3n(2m+3n)(2m-3n)m-n(n+m)(nm)2m3n(2m+3n)(2m3n)mn
(n+m)(n-m)2m-3n(2m+3n)(2m-3n)m-n(n+m)(nm)2m3n(2m+3n)(2m3n)mn
Step 4
Cancel the common factor of 2m-3n2m3n.
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Step 4.1
Factor 2m-3n2m3n out of (2m+3n)(2m-3n)(2m+3n)(2m3n).
(n+m)(n-m)2m-3n(2m-3n)(2m+3n)m-n(n+m)(nm)2m3n(2m3n)(2m+3n)mn
Step 4.2
Cancel the common factor.
(n+m)(n-m)2m-3n(2m-3n)(2m+3n)m-n
Step 4.3
Rewrite the expression.
(n+m)(n-m)2m+3nm-n
(n+m)(n-m)2m+3nm-n
Step 5
Expand (n+m)(n-m) using the FOIL Method.
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Step 5.1
Apply the distributive property.
(n(n-m)+m(n-m))2m+3nm-n
Step 5.2
Apply the distributive property.
(nn+n(-m)+m(n-m))2m+3nm-n
Step 5.3
Apply the distributive property.
(nn+n(-m)+mn+m(-m))2m+3nm-n
(nn+n(-m)+mn+m(-m))2m+3nm-n
Step 6
Simplify terms.
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Step 6.1
Combine the opposite terms in nn+n(-m)+mn+m(-m).
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Step 6.1.1
Reorder the factors in the terms n(-m) and mn.
(nn-mn+mn+m(-m))2m+3nm-n
Step 6.1.2
Add -mn and mn.
(nn+0+m(-m))2m+3nm-n
Step 6.1.3
Add nn and 0.
(nn+m(-m))2m+3nm-n
(nn+m(-m))2m+3nm-n
Step 6.2
Simplify each term.
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Step 6.2.1
Multiply n by n.
(n2+m(-m))2m+3nm-n
Step 6.2.2
Rewrite using the commutative property of multiplication.
(n2-mm)2m+3nm-n
Step 6.2.3
Multiply m by m by adding the exponents.
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Step 6.2.3.1
Move m.
(n2-(mm))2m+3nm-n
Step 6.2.3.2
Multiply m by m.
(n2-m2)2m+3nm-n
(n2-m2)2m+3nm-n
(n2-m2)2m+3nm-n
Step 6.3
Multiply n2-m2 by 2m+3nm-n.
(n2-m2)(2m+3n)m-n
(n2-m2)(2m+3n)m-n
Step 7
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=n and b=m.
(n+m)(n-m)(2m+3n)m-n
Step 8
Simplify terms.
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Step 8.1
Cancel the common factor of n-m and m-n.
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Step 8.1.1
Factor -1 out of n.
(n+m)(-1(-n)-m)(2m+3n)m-n
Step 8.1.2
Factor -1 out of -m.
(n+m)(-1(-n)-(m))(2m+3n)m-n
Step 8.1.3
Factor -1 out of -1(-n)-(m).
(n+m)(-1(-n+m))(2m+3n)m-n
Step 8.1.4
Reorder terms.
(n+m)(-1(m-n))(2m+3n)m-n
Step 8.1.5
Cancel the common factor.
(n+m)(-1(m-n))(2m+3n)m-n
Step 8.1.6
Divide ((n+m)(-1))(2m+3n) by 1.
((n+m)(-1))(2m+3n)
((n+m)(-1))(2m+3n)
Step 8.2
Simplify by multiplying through.
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Step 8.2.1
Apply the distributive property.
(n-1+m-1)(2m+3n)
Step 8.2.2
Reorder.
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Step 8.2.2.1
Move -1 to the left of n.
(-1n+m-1)(2m+3n)
Step 8.2.2.2
Move -1 to the left of m.
(-1n-1m)(2m+3n)
(-1n-1m)(2m+3n)
(-1n-1m)(2m+3n)
Step 8.3
Simplify each term.
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Step 8.3.1
Rewrite -1n as -n.
(-n-1m)(2m+3n)
Step 8.3.2
Rewrite -1m as -m.
(-n-m)(2m+3n)
(-n-m)(2m+3n)
(-n-m)(2m+3n)
Step 9
Expand (-n-m)(2m+3n) using the FOIL Method.
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Step 9.1
Apply the distributive property.
-n(2m+3n)-m(2m+3n)
Step 9.2
Apply the distributive property.
-n(2m)-n(3n)-m(2m+3n)
Step 9.3
Apply the distributive property.
-n(2m)-n(3n)-m(2m)-m(3n)
-n(2m)-n(3n)-m(2m)-m(3n)
Step 10
Simplify and combine like terms.
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Step 10.1
Simplify each term.
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Step 10.1.1
Rewrite using the commutative property of multiplication.
-12nm-n(3n)-m(2m)-m(3n)
Step 10.1.2
Multiply -1 by 2.
-2nm-n(3n)-m(2m)-m(3n)
Step 10.1.3
Rewrite using the commutative property of multiplication.
-2nm-13nn-m(2m)-m(3n)
Step 10.1.4
Multiply n by n by adding the exponents.
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Step 10.1.4.1
Move n.
-2nm-13(nn)-m(2m)-m(3n)
Step 10.1.4.2
Multiply n by n.
-2nm-13n2-m(2m)-m(3n)
-2nm-13n2-m(2m)-m(3n)
Step 10.1.5
Multiply -1 by 3.
-2nm-3n2-m(2m)-m(3n)
Step 10.1.6
Rewrite using the commutative property of multiplication.
-2nm-3n2-12mm-m(3n)
Step 10.1.7
Multiply m by m by adding the exponents.
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Step 10.1.7.1
Move m.
-2nm-3n2-12(mm)-m(3n)
Step 10.1.7.2
Multiply m by m.
-2nm-3n2-12m2-m(3n)
-2nm-3n2-12m2-m(3n)
Step 10.1.8
Multiply -1 by 2.
-2nm-3n2-2m2-m(3n)
Step 10.1.9
Rewrite using the commutative property of multiplication.
-2nm-3n2-2m2-13mn
Step 10.1.10
Multiply -1 by 3.
-2nm-3n2-2m2-3mn
-2nm-3n2-2m2-3mn
Step 10.2
Subtract 3mn from -2nm.
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Step 10.2.1
Move n.
-3n2-2m2-2mn-3mn
Step 10.2.2
Subtract 3mn from -2mn.
-3n2-2m2-5mn
-3n2-2m2-5mn
-3n2-2m2-5mn
Step 11
Move -3n2.
-2m2-5mn-3n2
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