Symbolab Algebra Cheat Sheet
Number Rules
• 𝑎⋅0=0
•
1⋅𝑎 =𝑎
Expand Rules
• −(𝑎 ± 𝑏) = −𝑎 ∓ 𝑏
• 𝑎 ⋅ (𝑏 + 𝑐) ⋅ (𝑑 + 𝑒) = 𝑎𝑏𝑑 + 𝑎𝑏𝑒 + 𝑎𝑐𝑑 + 𝑎𝑐𝑒
• −(−𝑎) = 𝑎
• 𝑎 ⋅ (𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐
• (𝑎 + 𝑏) ⋅ (𝑐 + 𝑑) = 𝑎𝑐 + 𝑎𝑑 + 𝑏𝑐 + 𝑏𝑑
Fraction Rules
0
• 𝑎 = 0, 𝑎 ≠ 0
•
𝑎
=1
•
(𝑏 )
•
𝑎−1 = 𝑎
•
𝑎
𝑎
𝑎 −𝑐
𝑐
𝑎 −1
•
(𝑏 )
•
𝑎−𝑏 = 𝑎𝑏
•
•
•
1
−𝑎
𝑏
𝑎
𝑏
( )
𝑐
1
𝑏
( )
𝑐
𝑏 𝑐
= ((𝑏) ) = (𝑎)
𝑎
•
= −𝑏
•
=
•
𝑎⋅𝑐
𝑏
𝑐
1
=𝑎
𝑎 −1
=
1
−𝑎
−𝑏
𝑎
−𝑏
𝑏
( )
𝑐
𝑎
1
𝑎
𝑏
( )
𝑏
=𝑎
𝑎
=𝑏
𝑎
= −𝑏
𝑏
= 𝑐⋅𝑎
=𝑏
Absolute Rules
• |−𝑎| = 𝑎, 0 ≤ 𝑎
• |−𝑎| = |𝑎|
•
•
|𝑎| = 𝑎, 0 ≤ 𝑎
|𝑎 ⋅ 𝑥| = 𝑎 ⋅ |𝑥|, 0 ≤ 𝑎
Exponent Rules
• 1𝑎 = 1
• 𝑎0 = 1, 𝑎 ≠ 0
• (𝑎𝑏)𝑛 = 𝑎𝑛 𝑏 𝑛
•
•
•
𝑛
•
•
•
•
𝑎𝑚
1
= 𝑎𝑛−𝑚 , 𝑚 < 𝑛
(𝑎𝑏 )𝑐 = 𝑎𝑏⋅𝑐
𝑎𝑛
𝑎 𝑐
𝑎𝑐
(𝑏 ) = 𝑏 𝑐
𝑎𝑐 ⋅ 𝑏 𝑐 = (𝑎 ⋅ 𝑏)𝑐
𝑛
𝑛
√𝑎 ⋅ 𝑏 = √𝑎 ⋅ √𝑏
𝑎1 = 𝑎
0𝑎 = 0, 𝑎 ≠ 0
•
𝑎𝑚
•
•
𝑎
= 𝑎𝑏 ⋅ 𝑎𝑐
𝑎𝑏⋅𝑐 = (𝑎𝑏 )𝑐
•
𝑎 𝑛 = ( √𝑎 )
= 𝑎𝑚−𝑛 , 𝑛 < 𝑚
𝑎𝑛
𝑏+𝑐
𝑚
𝑛
𝑚
Factor Rules
• 𝑥 2 − 𝑦 2 = (𝑥 − 𝑦) ⋅ (𝑥 + 𝑦)
• 𝑥 3 + 𝑦 3 = (𝑥 + 𝑦) ⋅ (𝑥 2 − 𝑥𝑦 + 𝑦 2 )
• 𝑥 𝑛 − 𝑦 𝑛 = (𝑥 − 𝑦) ⋅ (𝑥 𝑛−1 + 𝑥 𝑛−2 𝑦 + ⋯ + 𝑥𝑦 𝑛−2 + 𝑦 𝑛−1 )
• 𝑥 𝑛 + 𝑦 𝑛 = (𝑥 + 𝑦) ⋅ (𝑥 𝑛−1 − 𝑥 𝑛−2 𝑦 + ⋯ − 𝑥𝑦 𝑛−2 + 𝑦 𝑛−1 ), 𝑛 is odd
• 𝑎 ⋅ 𝑥 2⋅𝑛 − 𝑏 = (√𝑎 ⋅ 𝑥 𝑛 + √𝑏)(√𝑎 ⋅ 𝑥 𝑛 − √𝑏)
• 𝑎 ⋅ 𝑥 4 − 𝑏 = (√𝑎 ⋅ 𝑥 2 + √𝑏)(√𝑎 ⋅ 𝑥 2 − √𝑏)
• 𝑎 ⋅ 𝑥 2⋅𝑛 − 𝑏 ⋅ 𝑦 2⋅𝑚 = (√𝑎 ⋅ 𝑥 𝑛 + √𝑏 ⋅ 𝑦 𝑚 )(√𝑎 ⋅ 𝑥 𝑛 − √𝑏 ⋅ 𝑦 𝑚 )
• 𝑎 ⋅ 𝑥 4 − 𝑏 ⋅ 𝑦 4 = (√𝑎 ⋅ 𝑥 2 + √𝑏 ⋅ 𝑦 2 )(√𝑎 ⋅ 𝑥 2 − √𝑏 ⋅ 𝑦 2 )
Factorial Rules
• 0! = 1
• 𝑛! = 1 ⋅ 2 ⋯ (𝑛 − 1) ⋅ 𝑛
𝑛!
1
• (𝑛+𝑚)! = (𝑛+1)⋅(𝑛+2)⋯(𝑛+𝑚)
•
𝑛!
(𝑛−𝑚)!
= 𝑛 ⋅ (𝑛 − 1) ⋯ (𝑛 − 𝑚 + 1), 𝑚 < 𝑛
Log Rules
• log(0) =Undefined
• log 𝑎 (𝑎) = 1
1
• log 𝑎𝑏 (𝑥) = ⋅ log 𝑎 (𝑥)
•
𝑏
log 1 (𝑥) = − log 𝑎 (𝑥)
𝑎
1 𝑛
•
log 𝑥 ((𝑥) ) = −𝑛
•
log 𝑎 (𝑏) = ln(𝑎)
ln(𝑏)
Undefined
• 00 =Undefined
• log 𝑎 (𝑏) =Undefined, 𝑎 ≤ 0
• log1 (𝑎) =Undefined
Complex Number Rules
• 𝑖 2 = −1
• √−1 = 𝑖
• √−𝑎 = √−1 ⋅ √𝑎
•
•
•
log(1) = 0
log 𝑎 (𝑥 𝑏 ) = 𝑏 ⋅ log 𝑎 (𝑥)
1
log 𝑎 ( ) = − log 𝑎 (𝑥)
•
log (𝑥) =
𝑛
log 𝑥 (𝑥 𝑛 ) = 𝑛
𝑎log𝑎(𝑏) = 𝑏
•
•
•
•
𝑥
𝑥𝑛
𝑥
1
=Undefined
log 𝑎 (𝑏) =Undefined, 𝑏 ≤ 0
0