As a physics professor, I’ve discovered that the biggest impediment to success in college physics is a shaky foundation in high-school level algebra. If you are unsure of your algebra ability or want to review key concepts in preparation for physics, this page is for you.
Order of Operations
Consider the following mathematic expression:
x = [2(161/2+1) - 1]/3 - 9 - 32
To solve this expression requires a series of mathematical steps and it is imperative that one follows the order-of-operations when solving. The order-of-operations is a hierarchy describing the order in which mathematical operations must be performed. It is a standard that ensures that a mathematical expression always yields the same result. This hierarchy from first to last is as follows:
Parentheses/Brackets - Exponents/Roots - Multiplication/Division - Addition/Subtraction
Therefore, we first must pay attention to what is inside brackets and parentheses. Here there are two levels. So we first attack the terms inside the inner-most level of parentheses following the remainder of the order of operations. Thus, inside the parentheses, we must calculate the exponent of 1/2 (or square root) yielding:
x = [2(4+1) - 1]/3 - 9 - 32
No, we must perform the addition inside the inner-most parentheses yielding:
x = [2(5) - 1]/3 - 9 - 32
Now, we must attack the terms inside the brackets following the order or operations. Perfomring the multiplication first yields:
x = [10 - 1]/3 - 9 - 32
Next, the subtraction yields:
x = 9/3 - 9 - 32
Now, we have eliminated all of the parentheses and brackets so we can turn our attention to the remaining terms following the order of operations. We must next calculate the remaining exponent yielding:
x = 9/3 - 9 - 9
Now, perform the division yielding:
x = 3 - 9 - 9
Finally, perform the remaining subtractions yielding a result of:
x = -15
With more practice, you will learn (or remember) that several operations can be performed on one line of your paper, without violating the order of operations.
The order of operations also applies to arithmetic steps carried out when solving algebraic expressions except now the rules of algebra may limit the operations we can carry out. Keep the order of operations in mind but in general you can perform any operation to both sides of the equation so long as you apply the operation to all of the terms.
Consider the following algebraic expression:
(1x + 2x + 3)x = 27 + 3x
Let’s consider the terms inside the parentheses first. The like terms (in this case, terms that have an x) can be combined yielding:
(3x + 3)x = 27 + 3x
There is still more than one term inside the parentheses but we can not algebraically combine them since then do not both have an x in them. So we turn our attention to what is outside the parentheses and distribute the x on the left hand side by multiplying it with the terms remaining inside the parentheses.
3x2 + 3x = 27 + 3x
We could operate on the remaining exponent by taking the square root of both sides but we can forgo this for now without running into trouble. Let’s turn our attention to performing multiplications/divisions and come back to the square root later. Dividing both sides by 3 yields:
x2 + x = 9 + x
Now subtracting x from both sides yields:
Note that we could have first subtracted 3x from both sides then divided by 3 and arrived at the same result. Again, so long as an operation is applied to all terms on both sides of the equation you will not run into trouble.
Now we can solve for x by take the square root of both sides yielding:
x= +3 or -3
Algebra and Fractions
The single most challenging thing for my students to do correctly is algebra involving factions. Consider the following expression.
1 = 25/(x2)
For some reason, many students want to do a bone-headed thing like say that x2 = 1/25. This is very wrong. In order to solve an expression like this for x, we must first get x out of the denominator. Let’s multiply both sides by x2 yielding:
x2 = 25
Now, take the square root of both sides to solve for x yielding:
x= +5 or -5
A last word of advise: If this algebra review is Greek to you, then you are probably not ready for college physics. Take a algebra class or two before enrolling in physics or you are potentially setting yourself up for a harrowing experience.
IXL.com is an excellent resource to help you practice algebra.