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Dr. Math FAQ:
Browse High School Square & Cube Roots
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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
- 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
- Cube Root by Hand [01/23/1998]
How do you calculate the cube root of a number without using a
- 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
- 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
- Principal Square Root Positive [10/24/2002]
Why are we always taught that the principal square root of a number is
- 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) +
- 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?
- Solving for b [05/10/2000]
When solving 10 + 2*sqrt(b) = 0 we get b = 25, but when we insert 25 into
the equation, we end up with 20 = 0. What are we doing wrong?