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High School Archive

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.

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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?

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?

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)?

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 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?

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?

Simplifying Complex Numbers [05/23/2003]

Can you explain why the Product Rule doesn't apply to the problem sqrt(-49) x sqrt(-16) ?

Simplifying Square Root within Square Root without Calculator [07/05/2004]

I need to find sqrt[5 + sqrt(2)] without using a calculator. Is there a general formula for solving a problem like this?

Simplifying the Square Root of (a + b*sqrt(c)) [07/29/2004]

How do I find more advanced square roots, such as sqrt(11 - 2sqrt(18))?

Simplifying the Square Root of a Sum [09/27/2004]

Why is it that sqrt(a + b) does not equal sqrt(a) + sqrt(b)?

Solving Equations with Square Roots [03/28/2003]

When solving x + 1 = sqrt(x + 3), we can square both sides then solve the equation; however, we get an extra solution that doesn't work. Why?

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