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Explanation of allgebra terms: Allgebra is one of the most important sub-areas in mathematics. Simply put, allgebra means calculating with unknowns in equations.

Binomial formulas:

(a + b) ^ {2} = a ^ {2} + 2ab + b ^ {2} \

(a + b) ^ {3} = a ^ {3} + 3a ^ {2} b + 3ab ^ {2} + b ^ {3} \

a ^ {3} + b ^ {3} = (a + b) (a ^ {2} -ab + b ^ {2}) \

(a-b) ^ {2} = a ^ {2} -2ab + b ^ {2} \

(a-b) ^ {3} = a ^ {3} -3a ^ {2} b + 3ab ^ {2} -b ^ {3} \

a ^ {2} -b ^ {2} = (a + b) (a-b) \

a ^ {3} -b ^ {3} = (a-b) (a ^ {2} + ab + b ^ {2}) \

the Pascal triangle:

(a + b) ^ {3} & = & (a + b) \ cdot (a + b) \ cdot (a + b) & = & (a ^ {2} + 2ab + b ^ {2}) \ cdot (a + b) \ & = & a ^ {3} + a ^ {2} b + 2a ^ {2} b + 2ab ^ {2} + ab ^ {2} + b ^ {3} & = & a ^ {3} + 3a ^ {2} b + 3ab ^ {2} + b ^ {3} \

Potencies:

a ^ {0} = 1 \ quad, \ quad a ^ {1} = a \ quad, \ quad a ^ {2} = a \ cdot a \ quad, \ quad ... \ quad, \ quad a ^ { n} = \ underbrace {a \ cdot a \ cdots a} _ {{n \, {\ mathbf {mal}}}}

a ^ {5} = a \ cdot a \ cdot a \ cdot a \ cdot a \
For a> 0 \, a ^ {{- n}} = \ left ({\ frac {1} {a}} \ right) ^ {n} = {\ frac {1} {a ^ {n}} } \ and a ^ {{\ frac {m} {n}}} = {\ sqrt [{n}] {a ^ {m}}}, from which follows {\ sqrt [{n}] {a ^ {n }}} = a

Laws of calculation:

a ^ {r} \, a ^ {s} = a ^ {{r + s}}

{\ frac {a ^ {r}} {a ^ {s}}} = a ^ {{r-s}}

(a ^ {r}) ^ {s} = a ^ {{r \, s}}

(a \, b) ^ {n} = a ^ {n} \, b ^ {n}
\ left ({\ frac {a} {b}} \ right) ^ {n} = {\ frac {a ^ {n}} {b ^ {n}}
\ left ({\ sqrt [{n}] {a}} \ right) ^ {m} = {\ sqrt [{n}] {a ^ {m}}}

\ left ({\ sqrt [{n}] {a \, b}} \ right) ^ {m} = {\ sqrt [{n}] {a ^ {m}}} \, {\ sqrt [{n }] {b ^ {m}}}
\ left ({\ sqrt [{n}] {{\ frac {a} {b}}}} \ right) ^ {m} = {\ frac {{\ sqrt [{n}] {a ^ {m} }}} {{\ sqrt [{n}] {b ^ {m}}}}}

Logarithms

x = a ^ {y} \, \ Longrightarrow \, y = \ log _ {a} x
\ log (1) = 0 \,
\ log (x \, y) = \ log (x) + \ log (y)
\ log \ left ({\ frac {x} {y}} \ right) = \ log (x) - \ log (y)
\ log \ left ({\ frac {1} {x}} \ right) = - \ log (x)
\ log (x ^ {n}) = n \, \ log (x)
\ log {\ sqrt [{n}] {x}} = {\ frac {1} {n}} \, \ log (x)
\ log _ {a} x = {\ frac {\ log (x)} {\ log (a)}}
\ log _ {a} x = {\ frac {\ log (x)} {\ log (a)}}
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