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 TOPICS This page:   exponents    Search   Dr. Math See also the Internet Library:   exponents/roots HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Exponents Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Negative exponents.     Scientific notation. 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 only once. 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 notation. 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 that incorrect? Prime Numbers as the Difference of Two Squares [08/07/2002] Express the prime numbers 7, 15, and 261 as the difference of two squares. 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 irrational? 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 there? The Purpose of Powers [07/30/2002] I don't understand the purpose of powers, what they do, and how to use them. 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 them. 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 fraction? 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)* (4x^3)^(1/2)/[(27^4)^(1/3)]. 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) engineering notation. 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 to begin. Simplifying Expressions with Radicals [05/26/1998] Factoring to simplify square roots in expressions such as 12*sqrt(640)/3*sqrt(5). 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? Page: []

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