See also the
Dr. Math FAQ:
order of operations
Browse High School Basic Algebra
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Solving simple linear equations.
Positive/negative integer rules.
Completing the square.
Direct and indirect variation.
Inequalities and negative numbers.
- Natural Domain of a Function [03/12/1999]
Some inputs don't make sense for some functions.
- Natural Numbers Coprime to 6 [02/16/1998]
Let N(x) denote the number of natural numbers less than x which are
coprime to 6. Show that lim as x goes to infinity of [N(x)/x] = 1/3.
- Negative Exponents [10/23/1996]
I have to simplify this expression with non-negative exponents: x^5.
- Negative Numbers Combined with Exponentials [03/09/2001]
Why in the order of operations is negation a multiplication done after
exponentiation, rather than as a part of the base? What about
- Negatives and Inequalities [09/10/1998]
Why when we multiply or divide an inequality by a negative number do we
reverse the inequality sign?
- Negative vs. Subtraction in Order of Operations [10/30/2002]
I am trying to find some definitive source for treatment of leading
negative numbers in simplifying with order of operations.
- Nested Negative Exponents [8/2/1996]
How do I deal with nested negative exponents?
- Net Result of Consecutive Percentage Increases [12/11/2003]
Suppose a quantity is increased by some percentage, and then is later
increased again by another percentage. How can I calculate the
overall percentage increase?
- Nonlinear Factors [10/12/1998]
I have been told that factoring the sum of two squared numbers is not
possible, but factoring x^4 + 4 is possible. Please explain.
- Non-Linear Simultaneous Equations [12/07/2003]
I have four simultaneous equations with four variables, but each one
has a non-linear term in it (e.g., ab + c + d = 3). So none of the
usual techniques will work! How can I solve the system?
- Normalization [08/01/2001]
How do I figure out: 90 + 70 + 88 + 94 + x / 5 = 85 ?
- The Nth Root of N [11/28/2000]
Is the nth root of n (a whole number other than 1) ever a rational
- Number of Equations Needed in a Simultaneous Linear System [10/29/2003]
Could you tell me why we need the same number of equations as
variables in order to get a unique solution to a system of
simultaneous linear equations?
- Number * Sum of Remaining Four Numbers [04/03/2003]
Find 5 numbers such that when each number is multiplied by the sum of
the remaining 4 numbers, the following values will result: 152, 245,
297, 320, 360.
- Objects in a Pyramid [7/8/1996]
Objects are stacked in a triangular pyramid... how many objects are in
the nth layer from the top?
- Old Test Questions Answered [1/23/1995]
I am studying for my midterm, and I've come across two questions that I
got wrong and don't understand why....
- One Unknown, Decimal Power: By Inspection [10/18/2015]
An adult seeks to solve by hand for a variable raised to a decimal exponent. Doctor
Ali first simplifies the original equation to obtain its critical points and approximate its
maximum before re-introducing its factors and commenting on their effect.
- One Variable Equations with Decimals [02/11/1997]
How do you solve 8.25x + 3 = 17.5 + x?
- Open Sentence, Statement [09/18/2001]
What is an open sentence?
- Operator Precedence [08/13/2003]
Since the following statement is true: (1+1)**(5-2) is 8, why is the
following statement true and not false: 2**1+1 is 3 and not 4, and
3*1**3 is 3 and not 27...' ?
- Ordering Exponents and Variables [04/08/2000]
Is there a rule for putting terms in descending order if the variables
have the same exponent? What about negative exponents and descending
- Ordering Products, Powers, and Parameters of Trigonometric Functions [10/31/2010]
A student wants to know how to unambiguously interpret strings of trigonometric
functions, multiplication, and exponentiation. Doctor Peterson digs into a history book
-- as well as another math doctor's conversation -- to illuminate the vagaries of the
- Order in Linear Expressions [11/20/2001]
Can you give me a convincing argument as to why, in an equation such as
y=13-7x, where y is a function of x, it should be written as y=-7x+13?
- Order of Operations [05/19/1999]
Given a, b, x, and y, find ax/by.
- Order of Operations with Percentages [04/05/2001]
Why does the order of operations exclude percentage, square roots, etc.?
- Other Ways to Get the Quadratic Formula [02/19/2010]
Doctor Jacques shares two ways to derive [-b +/- SQRT(b^2 - 4ac)]/2a
without explicitly completing the square.
- Pairs of Odd Integers [09/24/1997]
Find all pairs of odd integers a and b that satisfy the equation: a +
128b = 3ab.
- Parabola with Horizontal or Vertical Axis [05/03/2001]
How can I find the focus and directrix of a parabola whose axis is either
horizontal or vertical, like 2x^2 = -y?
- Parabolic Golf Shot Equations [01/24/2002]
Does the ball reach the green?
- Parallel and Perpendicular Lines [01/14/1999]
How do you tell without graphing whether the graphs of these equations
are parallel, perpendicular, or neither?
- Parallel Lines [12/31/1998]
What are some ways of proving lines parallel - geometrically and
- Parametric Form for Equation of a Line [6/30/1996]
How can you convert an equation such as y = -3x/4 + 7/2 to parametric
- Parity [8/2/1996]
How do I determine if the given function is odd, even or neither? What's
the symmetry of the graph?
- Partial Fractions [01/29/1998]
How do I express 3/1-(x^3) in partial fractions?
- Pascal's Triangle Pattern [04/22/1999]
What pattern does multiplying each entry by 1, 2, 3, 4, 5... in order,
and adding the products yield?
- Perfect Square: Solving Two Equations [6/14/1996]
x^2 + 5 is a perfect square, and x^2 - 5 is a perfect square.
- Perimeter Equals Area in a Triangle [4/2/1996]
When will the area and perimeter of a right triangle be numerically
- Picture Frame, Triangle Measurements [5/20/1996]
My teacher gave us ten questions to answer and I could do all except two:
1) A framed rectangular picture is 35cm long and 25cm wide... 2) The base
of a triangle is 9cm more than the perpendicular height...
- Plus or Minus Sign [03/08/2002]
What does this equation mean: y = +- k ? The - sign is directly under the
- Point Equidistant from 3 Other Points [04/11/1999]
How do you find a point that is equidistant from three other points?