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Selected answers to common questions:
- Newton's Method and Square Roots [12/04/1998]
Can you explain how finding square roots by hand relates to Newton's
method for approximating the zero of a function?
- n Factorial - Prove Lower Bound is n^(n/2) [09/11/2001]
I am trying to prove the following: n^(n/2) <= n!
- Non-integer Powers and Exponents [01/06/1999]
How do you find x^n, where n can be an integer, a fraction, a decimal, or
an irrational number?
- The N'th Root [07/26/1997]
How can you find the n'th root without a calculator?
- Number and Its Square Using All 9 Digits Exactly Once [05/22/1998]
Using multiplication facts to find all the whole numbers for which the
number and its square together use exactly nine digits 1, 2, 3, ..., 9
- Number of Ancestors 1000 Years Ago [11/22/1999]
How many living relatives would I have had 1,000 years ago, assuming 25
years per generation?
- Number of Roots of Polynomial with Radical [02/05/2002]
Solve this equation:r-9(r^(1/2))+8 = 0.
- Numbers Raised to the Negative Power [11/14/2001]
I know that 5^(-N) = 1/5^N. I would like to know why.
- 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
- Perfect Squares and Inspection [5/23/1995]
If x^1/2 + y =11 and y^1/2 + x = 7, what are x and y?
- Perfect Squares and Irrational Numbers [02/13/2002]
Isn't any non-perfect square an irrational number? What is the number
0.49? Its square root is 0.7, which is neither irrational nor an integer.
- Pi-th Root of -1 [12/15/2000]
How can you find the pi-th root of -1?
- Polynomial Function and Exponential Growth [06/08/1999]
How can I find a polynomial function with integer coefficients that has
3, 2, -4 and -1 as zeroes? What is the population of a colony of bacteria
20 seconds after it is discovered?
- Powerful Comparison [12/15/2014]
A teen wonders how to compare 21^8 and 5^15 by hand. Picking up on a technique
that the student knows and extending it into inequalities, Doctor Douglas judiciously
chooses some nearby decimal fractions.
- Power or Exponent? [10/14/2003]
I always thought that 'power' was just a synonym for 'exponent'. Is
- Prime Numbers as the Difference of Two Squares [08/07/2002]
Express the prime numbers 7, 15, and 261 as the difference of two
- Proof that Sqrt(3) is Irrational [08/14/1997]
How does one prove that sqrt(3) is irrational? or others? Is there a
general algorithm? How about just for primes?
- Prove or Disprove that Sqrt(2) and Sqrt(4) are Irrational [6/23/1995]
I have learned the following proof that sqrt(2) is irrational...If you
use this proof with sqrt(4) you can proof that sqrt(4) is irrational!!!
Where did I go wrong?
- Prove that Log A is Irrational [06/14/1998]
Can you help me prove that a common log of a number (not powers of 10) is
- Proving a^x = a^y iff x = y [12/13/2000]
How can I prove that a^x = a^y iff y = x for all real numbers x and y?
- Proving e is Irrational [11/19/1997]
My professor suggested using a proof by contradiction, but I don't
understand how to do it.
- Proving the Square Root of 3 is Irrational [12/7/1995]
I have to prove that the square root of 3 is irrational.... First we must
assume that: sqrt 3 = p/q; then I have: 3 = p^2/q^2. Where do I go from
- The Purpose of Powers [07/30/2002]
I don't understand the purpose of powers, what they do, and how to use
- Raising a Negative to an Irrational Power? It Depends [03/16/2011]
Why can't you evaluate a negative number raised to an irrational power? Doctor Vogler
explains how this depends on definitions of exponentiation, then describes three of
- Raising Sqrt(2) [10/14/1997]
What is the value of x in the following equation: x^x^x^x^x ... = 2 ?
- Rationalizing a Cubic Root Denominator [12/06/1997]
I am trying to rationalize the equation: 1/cubrt(9)-2.
- Rationalizing the Denominator [07/20/2001]
Is it true that you must get rid of square roots in the denominator of a
- Resolving Decimal Exponents [03/26/2001]
How can you find 7^.3 or 5^.6 without using a calculator? Is there more
than one way of doing this?
- Rule of 72 [01/26/1999]
Why does the rule of 72 work? For what interest values does it work best?
- Rules of Exponent Manipulation [1/22/1996]
f sub r equals the reciprocal of ...
- Rules of Exponents [10/02/1999]
Use the rules of indices to simplify the expression [(16x^2)^(1/4)*
- Scientific Notation: Adding and Subtracting Exponents [03/21/2002]
How do you add and subtract numbers written in scientific notation form?
- Scientific Notation and Engineering Notation [10/10/2001]
Convert 0.000 000 000 000 000 000 000 3 to: a) scientific notation; b)
- Shortcut for Computing x^n [07/05/2002]
Is there a shortcut that can be used to compute values like 3^12,
without doing 12 separate multiplications?
- Show that x = 2 [08/29/2001]
Given that: x=(10+(108)^(1/2))^(1/3)+(10-(108)^(1/2))^(1/3), show that x
= 2 without using a calculator.
- Significant Figures and Scientific Notation [05/21/1997]
How do you tell how many significant figures a number has?
- Simplifying Cube Roots in the Denominator [07/12/2001]
Simplify: the cube root of 3 all over the (cube root of 2) minus the
(cube root of 4).
- Simplifying Exponential Equations [08/19/2003]
A review question: (3r^-1s^2t^3)^2/27r^3st^-2. I don't even know how
- Simplifying Expressions with Radicals [05/26/1998]
Factoring to simplify square roots in expressions such as
- Simplifying Radicals [06/18/2001]
In the following: sin(15) = sqrt(2-sqrt(3))/2 = (sqrt(6)-sqrt(2))/4, how
do you manipulate sqrt(2-sqrt(3))/2 to obtain (sqrt(6)-sqrt(2))/4?