Order of Operations

The “Order of Operations” is a convention that ensures everyone interprets algebraic notation the same way. It is used in conjunction with the properties of each operation (addition, subtraction, multiplication, division, and exponentiation). I summarize the order of operations for myself as “do the most powerful operations first”.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

3+2(15 \div 3-1)^2

=5(15 \div 3-1)^2

=(75 \div 15-5)^2

=(5-5)^2

=0^2

=0

.

..

Hint 1: The correct answer to the original problem is: 35

Hint 2: What if the problem had been written
3+2\cdot (15 \div 3-1)^2

Hint 3: Two mistakes have been made in the work above

 

Sums and Products with Exponents

Powers and roots can be distributed over products and quotients.  They may not be distributed over sums or differences, no matter how tempting it may be. Sums or differences raised to a power must be used as a factor the indicated number of times, then multiplied using the distributive property.

An exponent applies only to the factor immediately below it unless parentheses have been used to indicate otherwise.

“Like terms” have the same variables, to the same powers… and only “like” terms may be combined by adding their coefficients.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

(f-g^2)^2-fg^2-(-fg)^2-f^2

=f^2-g^4-fg^2-(-fg)^2-f^2

=f^2-g^4-f^2g^2-(-fg)^2-f^2

=f^2-g^4-f^2g^2+fg^2-f^2

=-g^4

.

..

Hint 1: The correct answer to the problem on the first line above is: g^4-3fg^2-f^2g^2

Hint 2: The work shown above contains sign errors, simplification errors, and exponentiation errors. Re-read the text at the top of the posting if you have not found them all…

Negative Exponents

Negative exponents can be another source of confusion. No matter where you find a negative exponent, you can turn it into a positive exponent by taking the reciprocal of the expression it applies to.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

\dfrac{c^{-2}d^3}{c^3d^{-4}}

=\dfrac{d^3d^4}{c^{-2}c^3}

=\dfrac{d^7}{c^{-6}}

=d

.

..

Hint 1: The correct answer to the original problem is: \dfrac{d^7}{c^5}

Hint 2: Several mistakes have been made in the work above – don’t just seek the correct answer – look for the mistakes between every pair of lines.

Laws of Exponents

The “laws of exponents” are a frequent source of errors.  The rules that apply when simplifying expressions that involve exponents can be figured out quickly on your own if you (in your mind’s eye) expand integral exponents into repeated multiplication. I encourage students to master being able to explain why each of these rules is as it is instead of memorizing them, as memorized versions are more likely to get jumbled together in your thinking when working problems.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

\dfrac{(ab^2)^3(5a^2)}{a^2b}

=\dfrac{a^5b^525a^2}{a^2b}

=\dfrac{25a^{10}b^5}{a^2b}

=25a^5b^5.

.

..

Hint 1: The correct answer to the original problem is: 5a^3 b^5

Hint 2: Several mistakes have been made in the work above – don’t just seek the correct answer – look for the mistakes between every pair of lines.