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Browse High School Square & Cube Roots
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Square roots without a calculator.
    Table of squares & square roots, 1-100.

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

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

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

Bakhshali Formula [12/15/2002]
I came across this formula for calculating square roots by hand.

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

Calculating Any Root [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...

Calculators and Irrational Numbers [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?

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

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

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

Cube Root Calculation, Explained [04/18/2002]
It was good to see the way you outlined to calculate the cube root manually, but I wasn't able to understand.

Cube roots [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).

Definition of Negative Square Roots [03/08/2004]
I know that the square root of 49 = 7 since 7 x 7 = 49. But the negative square root of 49 is -7. Is this because (-7) x (-7) also equals 49 or because the square root of 49 is 7 and the negative stays because it is not involved with the operation? My teacher wrote -SQRT(49) = -7 because (-7) x (-7) = 49.

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

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

Exact Answers for Square Roots [04/16/2006]
My teacher says that one of the problems with using calculators is that you don't always get exact answers. I don't understand why. Aren't calculators accurate?

Factoring out a Fraction's Fifth Roots [10/17/2011]
A student struggles to rationalize a fraction with fifth roots in the denominator. Doctor Vogler furnishes the necessary complex conjugates after showing how they work first with square roots, then with cube roots, and finally with fourth and fifth roots.

Fractional Exponents [12/07/2003]
When a number is raised to a power like 4/3 or 3/5, how is it done?

History of the Root of an Equation [11/01/2007]
Why are solutions to equations referred to as roots?

How Do Cube Roots Work? [11/04/2003]
I understand square roots but I'm not sure how to do cube roots. Can you help?

Is 14798678562 or 15763530163289 a Perfect Square? [12/08/2002]
Examine both the units digits and the digital roots of perfect squares to help determine whether or not a given number is a perfect square.

Is the Square Root of i^4 Equal to 1 or -1? [02/24/2004]
If you take the square root of i to the fourth power, does that equal i to the second power, which is equivalent to -1? Or can you simplify under the radical first and say i to the fourth power is 1 and the square root is then 1? Which approach is correct?

Mental Math Tricks: Finding Cube Roots of Large Numbers [06/07/2004]
A friend asked me to pick a number between 100 and 200, cube it, and give him the answer. After thinking about it, he gave me the original number that I had cubed--the cube root of the number I gave him. How does he do this in his head without a calculator?

Mental Math Tricks - Finding Two Digit Square Roots [12/03/2003]
Do you know if there is any trick behind looking at a number which is the result of a two digit number having been squared and being able to tell what the number was that was squared?

More Methodical Than Guessing [06/12/2017]
A young adult who prefers clear procedures struggles with guessing to calculate square roots and factor polynomials. Distinguishing between operations and algorithms, Doctor Peterson surveys a range of methods for finding roots and factoring — some of which require no guessing, others of which involve approximating — before emphasizing the intuition that develops with perseverance.

Multiplying Square Roots of Negative Numbers [11/25/2003]
It seems to me there are two possible ways to interpret a problem like sqrt(-2) * sqrt(-2). One way I get 2 and the other way I get -2. Which solution is correct? What's wrong with the other one?

Newton's Method and Continued Fractions [10/06/1999]
Can you clarify some points on Newton's method of finding square roots without a calculator, and on the continued fraction algorithm (CFA)?

One Variable in Two Radicals [05/15/2017]
A young adult struggles to solve for a variable that appears under two separate square root expressions. Doctor Peterson guides her back on her way, to much mutual satisfaction.

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

Prime Factors and Square Products [10/05/2003]
What is the smallest number that you can multiply by 540 to make a square number?

Principal Square Root Positive [10/24/2002]
Why are we always taught that the principal square root of a number is positive?

Proving Two Radical Expressions Are Equivalent [06/28/2005]
I calculated an answer to a given problem and got a radical expression. In attempting to confirm my answer on the Internet, I found a different radical expression. Though my calculator suggests that the decimal forms of the two are equivalent, I have been unable to algebraically manipulate them and show that. Can you help?

Rationalizing a Denominator with Multiple Cube Roots [04/22/2011]
A student of field theory wonders how to remove the cube roots from the denominator of 1/(a + b*CBRT(q) + c*CBRT(q)^2). Building on the conjugacy of square roots, Doctor Vogler writes out the required conjugates.

Rationalizing Denominators with Multiple Radicals [11/04/2004]
How can I rationalize the denominator of a fraction when it contains many square roots, such as 1/[sqrt(3) + sqrt(5) + sqrt(7) + sqrt(11) + sqrt(13)] ?

Rationalizing Roots in More Denominators [02/16/2011]
A student wonders if there is a systematic approach to rationalizing denominators that contain different roots -- or if it's even possible without getting into roots of unity. Doctor Vogler outlines the method for simple conjugates and more involved ones.

Rationalizing the Denominator [07/10/2003]
1 / ((sqrt)3 + (sqrt)5 + (sqrt)7)

Restrictions on Roots [06/20/2002]
When we talk about the nth root of a number, are there any restrictions on n, other than that it can't be zero?

A Sequence of Square Roots, Nested — and Bounded from Above [08/07/2012]
Doctor Vogler proves that an upper bound exists for the limit of the sequence 1, sqrt (2), sqrt(2sqrt(3)), sqrt(2sqrt(3sqrt(4))), sqrt(2sqrt(3sqrt(4sqrt(5)))) ...

Simple Number Pair Series Yields Surprising Ratio ... Why? [12/31/2009]
An enthusiast wonders about the curious ratio that emerges from a simple pattern for generating number pairs. Doctor Rick builds an algebraic argument for why its phi-like recursive relationship approaches the square root of 2.

Simplifying and Working with Imaginary Numbers [04/11/2008]
What is the rule for simplifying an expression like sqrt(50)/sqrt(-5)? Do you get i*sqrt(10) or -i*sqrt(10)? Is there a general rule for simplifying imaginary square roots with regard to handling the i?

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