Square and Cube Roots

• Adding Square Roots [Ollie, 12/26/2001]
  What is the square root of 3 + the square root of 27?

• Cube Root by Hand [Van Cauwenberghe, 01/23/1998]
  How do you calculate the cube root of a number without using a calculator?

• Cube roots [Elena, 05/30/1997]
  How can I can figure out the cube root of a number? (i.e., that the cube root of 216 is 6).

• Do Irrational Roots Always Occur in Conjugate Pairs? [Libby, 03/13/2002]
  A colleague and I have been puzzled by the equation x^3+4x^2+3x+1=0. Upon graphing, there appear to be two imaginary solutions and one real solution...

• Finding Square Roots [Mardekian, 09/06/1998]
  Can you explain why the algorithm for finding square roots works, without using algebra?

• Heron's Method for Finding Square Roots by Hand [Alvin, 12/18/2001]
  How do you get the square root of a number manually without a table or calculator?

• Longhand Square Roots [Tim, 03/30/1998]
  How do you find the square root of any number? Is there an easy formula?

• Negative Square Roots on Calculators [Graciela, 10/30/1995]
  Every time I try to calculate on my calculator the square root of a negative number (-4, for example) I get the message "Error". Why?

• Palindromic Squares [Jimmy, 07/28/1997]
  Do you know any numbers besides 14641 where both the number and its square root read the same left to right as right to left?

• Proof that Sqrt(2) is Irrational [Tim, 8/29/1996]
  What is the proof that the square root of 2 is irrational? I recall it begins with the assumption that sqrt(2) is rational...

• Square Root of a Negative Number [Pedlar, 01/25/1997]
  Is it possible to find the square root of a negative number and, if so, to what number system do these square roots belong?

• Why Simplify Square Roots? [Cendrowski, 05/17/1999]
  We have been trained to simplify square roots, but we are now looking at the reasons why we should continue to teach the process.

• Perfect Squares and Inspection [Muhtar, 5/23/1995]
  If x^1/2 + y =11 and y^1/2 + x = 7, what are x and y?

• Prove or Disprove that Sqrt(2) and Sqrt(4) are Irrational [Rubitschka, 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?

• Simplifying Radicals and Exponents [Seguin, 9/3/1995]
  A student asks how to simplify square roots and numbers with exponents.

• Simplifying Square Roots [Seguin, 9/5/1995]
  Could you help me with this problem? 1/2 *Sqrt{24}

• Adding Square Roots [Looney, 10/31/1995]
  I don't understand how to add 3 Radical 81 + 2 Radical 25.

• Proving the Square Root of 3 is Irrational [Meyer, 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?

• Conjugates with Square Roots [Fowler, 1/29/1996]
  I don't understand what conjugates are in relation to square roots, and what good they do.

• Functionality of the Square Root [Briem, 4/27/1996]
  I maintain that the relation indicated by the statement: SQRT(x) = 3D y (limited to real numbers) is a function, whereas my teacher claims it is not.

• Square root of a complex number. [Lindsey, 6/6/1996]
  Find the square root of 5+12i. (Let the square root of 5+12i = a+bi.)

• Square Roots Without a Calculator [David, 8/6/1996]
  How do you get a square root without a calculator? What is the square root of 7 to 5 decimal places?

• Square roots by hand: Divide and Average [Rodgers, 8/31/1996]
  How do you calculate square roots by hand?

• Why Multiple Roots? [Duff, 9/9/1996]
  Why does raising an equation to an even power introduce an extraneous root?

• Descartes' Square Root Method [Potivn, 10/30/1996]
  Why does Rene Decartes' geometric method for finding square roots work?

• Simplifying with Cube Roots [Grossman, 01/22/1997]
  Simplify: cube root(3) / [cube root(6) + cube root(3)]

• Square Root of a Negative Number Squared [Smith, 01/23/1997]
  Is Sqrt(-6)^2 equal to 6 or -6?

• Cube Root Algorithm [Walters, 04/04/1997]
  Is there an alorithm for working out the cube root of numbers without a calculator?

• History of the Radical Sign [Younger, 04/17/1997]
  Who invented the radical sign?

• The N'th Root [Brown, 07/26/1997]
  How can you find the n'th root without a calculator?

• Square Root Trick [Lee, 08/13/1997]
  Is there a formula to calculate sqrt(x+sqrt(x+sqrt(x+... ?

• Proof that Sqrt(3) is Irrational [Gardner, 08/14/1997]
  How does one prove that sqrt(3) is irrational? or others? Is there a general algorithm? How about just for primes?

• Calculating Any Root [Tat, 10/13/1997]
  I need to find an algorithm to determine any root of a number. I was told I could determine the estimated value by using Newton's Method...

• Raising Sqrt(2) [Benno, 10/14/1997]
  What is the value of x in the following equation: x^x^x^x^x ... = 2 ?

• Cubic Functions [Hays, 10/15/1997]
  How do I solve a cubic equation like f(x) = x^3-2x^2+4x-5 ?

• Cube Roots of Numbers [Martin, 11/05/1997]
  If you take i (sqrt(-1)), the cube root is -i, but since x^3 = i is degree three there should be three different values of x. What are they?

• Square Roots in Complex Numbers [Jurd, 11/06/1997]
  Why in the complex number system does every number have two square roots, when in the real number system we teach that the square root of any positive number is by definition POSITIVE?

• Rationalizing a Cubic Root Denominator [Walker, 12/06/1997]
  I am trying to rationalize the equation: 1/cubrt(9)-2.

• Cubic Equations [Colin, 01/19/1998]
  How would you solve a problem like 3x^3+x^2+15x+27 = 0?

• Indirect Proof [Hanson, 02/14/1998]
  Prove that sqrt(2) is an irrational number.

• Four Methods for Extracting Cube Roots [Rizwan, 03/03/1998]
  Extracting cube roots by factoring, calculator, guess and check, and iteration.

• Computing Square Roots Manually [Ross, 03/05/1998]
  Using the bisection method to compute square roots manually.

• Theorem About Sum and Product of Quadratic Roots [laura, 03/07/1998]
  Explain why 1/2 and 3/4 are NOT the roots of 0 = 4x^2 + 5x + 8.

• Deriving a Formula for Cubic Roots [Renton, 05/22/1998]
  Is there a working formula to solve the 3 roots of a cubic? I'm trying to derive it but am having trouble...

• Simplifying Expressions with Radicals [Mallorie, 05/26/1998]
  Factoring to simplify square roots in expressions such as 12*sqrt(640)/3*sqrt(5).

• Solving Radical Expressions [Sherri, 07/26/1998]
  How do you solve radical expressions such as the cube root of 32 minus the cube root of 108?

• Using Conjugates to Simplify Radicals [Varun, 08/04/1998]
  When using conjugates, how do we know when to change the sign? For example, how would we simplify 1/[3+sqrt(3)-sqrt(6)]?

• Newton's Method and Square Roots [Hendricks, 12/04/1998]
  Can you explain how finding square roots by hand relates to Newton's method for approximating the zero of a function?

• Square Root Function [Lynch, 12/16/1998]
  What is the difference between finding the solutions to x^2 = 4 and finding sqrt(4)? Why does sqrt(2x-3) = -1 have no solutions?

• Dividing Radicals [Holly, 02/15/1999]
  How do you simplify 7 sqrt32 / (5 sqrt63) ?

• Square Roots Of Complex Numbers [Max, 02/22/1999]
  Find the square roots of 5-12i.

• Dividing and Multiplying Radicals [Marta, 03/07/1999]
  How do you multiply the square root of 3/4 by the square root of 4/5?

• Fractional Exponents [Tom, 03/11/1999]
  I am trying to figure out the water consumption of the average goat.

• Square Root of 100 [Cry, 04/15/1999]
  Why can't -10 be a square root of 100?

• Calculating 4th Roots, 5th Roots... [Smith, 05/05/1999]
  Is there an easy way to calculate roots of any given depth?

• Sqrt2 Irrational [Arjun, 05/17/1999]
  How do you prove that the square root of 2 (or other roots of other real numbers) is irrational?

• Square Root of 3 [Sheila, 05/18/1999]
  How can I prove that the square root of 3 is an irrational number?

• Derivation of the Equation of an Ellipse [Walsman, 06/23/1999]
  How can I derive the equation of an ellipse from its definition?

• Surds [Flaspohler, 05/17/2000]
  Where does the term "surd" for square roots come from?

• Square Roots in Binary [Watkins, 10/03/2000]
  Can you show an example of taking the square root of a binary number?

• Un-nesting Radicals [Tran, 03/16/2001]
  How can I simplify sqrt(2+sqrt[3])?

• Calculators and Irrational Numbers [King, 05/02/2001]
  When I square the square root of 11 on any calculator, I get the answer 11 (exactly). That seems to indicate that the square root of 11 is a rational number, but it's not. Can you explain this?

• Extraneous Roots [Kostya, 06/13/2001]
  The solution 4 (to sqrt(x-3) = x-5) only works for a negative root; 7 only works for a positive root. Which solution is valid?

• Simplifying Radicals [Alec, 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?

• Investigation Involving Square Root of 2 [Loka, 07/09/2001]
  How can you explain the fact that (665857/470832)^2 = 2?

• Simplifying Cube Roots in the Denominator [Pauline, 07/12/2001]
  Simplify: the cube root of 3 all over the (cube root of 2) minus the (cube root of 4).

• Rationalizing the Denominator [Matt, 07/20/2001]
  Is it true that you must get rid of square roots in the denominator of a fraction?

• Formula for Mth Root [Simon, 08/16/2001]
  Since the square root of n is the same as n^(1/2), what about n^(1/3) or n^(1/m) or m to the inverted mth power? Is there a formula?

• Show that x = 2 [Andrew, 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.

• Solving Third-Degree Equations [Jan, 09/24/2001]
  I'm trying to solve x^3 + 2x^2 - 3x - 4 = 0. I need to take the square root of 4p^3 + 27q^2, which is -272. I'm looking for the real solutions, and there are three of them.

• Finding Roots of Polynomials with Complex Numbers [Ed, 09/27/2001]
  I read in the archives that you can find the roots of 3rd or higher- degree polynomials with complex numbers...

• Equation without a Solution [Steven, 11/14/2001]
  What is the solution to the equation sqrt(x) = -2 ?

• Cubic Equations in One Formula [Robert, 11/27/2001]
  I know the formula for solving quadratic equations of the type ax^2+bx+c=0, but is there a formula for cubic equations?

• Complex Radicals [Anna, 11/27/2001]
  I understand the square root, but how do you use the cubed root?

• Etymology of the Word Radical [Chrissy, 12/28/2001]
  Where does the word radical come from?

• Number of Roots of Polynomial with Radical [Meghan, 02/05/2002]
  Solve this equation:r-9(r^(1/2))+8 = 0.

• Perfect Squares and Irrational Numbers [Mimi, 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.

• Origin of the Word Root [Naomi, 02/18/2002]
  If we set an expression equal to zero, we call the solution the "root" of the equation. Why?

• Nested Radical [Mac, 02/25/2002]
  Prove the following nested radical: sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...)))) = 3

• Square Roots using Basic Arithmatic [Tony, 03/19/2002]
  Given X and Y, solve for Z using only basic operations: X = Z^Y; X = 729, Y = 6.