Student Misconceptions

Common mistakes that students make ..
Multiplying base and exponent     

     Incorrect:    2³ ≠ 2 x 3 

     Correct:    2³ = 2 x 2 x 2 = 8   

Exponents "distribute"
     Incorrect:  (2 + 4)³ ≠ 2³ + 4³
     Correct:    (2 + 4)³ = 6³ = 6 x 6 x 6 =216

Anything to the power of "zero" is? ...
     Incorrect:  7⁰ ≠ 0 
     Correct:    7⁰ = 1  
     Proof:       7⁰ = 7^(m-m) = 7^m ÷ 7^m = 1 (anything divided by itself is just "1") 

Scientific Notation! A negative on an exponent and a negative on a number ... uhh? Are they the same? 

     Correct:  0.00036 = 3.6 × 10^-4 

                 ^{-0.00036 = -3.6 × 10-4}                  

                 ^{36,000 = 3.6 × 104}

                 ^{-36,000 = -3.6 × 104}

A negative on an exponent and a negative on a number are equal 

     Incorrect:   -4² ≠ 16 

     Correct:    -4² = -16

     Correct:    -4² = -(4)² = -16 

     Correct:    (-4)² = 16 

How to fix this? 

A common misconception that can occur with instances 3 is that students believe that since

(3 x 2)² = 3² x 2² then (3 + 2)² = 3² + 2², or the exponent rules when multiplying numbers with exponents is applicable to all operations.

Perhaps further examples can be used to illustrate this point:

i.e. (pq)³ = (pq) x (pq) x (pq) = (p x p x p) x (q x q x q) = p³ x q³

(p + q)³ = (p + q) x (p + q) x (p + q) = use of distributive property to solve.

i.e. (3 x 4)³ = (3 x 4) x (3 x 4) x (3 x 4) = (3 x 3 x 3) x (4 x 4 x 4) = 3³ x 4³, which is same as 12³   but

i.e. (3 + 4)³ = (3 + 4) x (3 + 4) x (3 +4) = 7³, which is not the same as 3³ + 4³