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Dr. Math FAQ:
Browse High School Square & Cube Roots
Stars indicate particularly interesting answers or
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Selected answers to common questions:
Square roots without a calculator.
Table of squares & square roots, 1-100.
- 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)?
- sin(sqrt2) More Simply? [05/17/2017]
A teen seeks simpler forms for trigonometric functions of radicals. With examples,
Doctor Vogler explains how this pursuit leads to infinite cardinalities of sets and further number theory.
- 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?
- Sqrt (x-2) = Sqrt (x+2) ? [01/10/2003]
Is the following equation solvable for x? sqrt (x-2) = sqrt (x+2)
- Square Root Function [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?
- Square Root Function: Why Restrict Its Range to Non-Negative Numbers? [01/26/2010]
Doctor Peterson makes the case for non-negative principal roots.
- Square Root of 100 [04/15/1999]
Why can't -10 be a square root of 100?
- Square Root of a Negative Number Squared [01/23/1997]
Is Sqrt(-6)^2 equal to 6 or -6?
- Square Roots and Limits [03/12/2003]
Given a value of x, find the value of the expression sqrt(x sqrt(x
- Square roots by hand: Divide and Average [8/31/1996]
How do you calculate square roots by hand?
- Square Roots, Complex Numbers [6/15/1996]
x^2 = -9 : I tried taking the square root of both sides. Is this
- Square Roots in Complex Numbers [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?
- Square Roots Without a Calculator [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 Root Symbol [04/29/2002]
Why is there a 2 in the index of a radical?
- Subtleties of Radical Equations [12/07/2003]
Can you explain how to solve radical equations, like sqrt(5x) = 1.5?
- Sum Square Root Search [03/06/2015]
An adult wonders for what integer is the sum of the digits of its square equal to the
square root of the number. With some programming premised on logarithmic thinking,
Doctor Ali provides numerical solutions.
- Two New Variables [10/25/2002]
I am trying to calculate using the Owens-Wendt equation. The equation
will give two equations with two unknowns. I am unable to solve
them, because of the x^1/2 and y^1/2 powers.
- Using Conjugates to Simplify Radicals [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)]?
- The Value of ABS[?] [08/24/2010]
Absolute value symbols keep appearing out of nowhere. Doctor Ian demystifies.
- A Way to Estimate the Square Root of Any Number [03/16/2006]
A student studying square roots discovers an interesting method of
estimating non-perfect roots and is curious why it works.
- What Are the Definitions of Zero and Root? [12/23/2003]
What is a zero, and what is a root? How are they used with regard to
equations? Different sources seem to be giving me different answers,
some of which contradict each other. What actually IS a root or zero?
- What's the Square Root of 75? [08/18/2002]
Is the square root of 75 15(sqrt 5) or 5(sqrt 15)?
- Why Does the Order of Square/Square Root Matter? [8/28/1996]
Why does the square root of x^2 equal absolute value of x, but the square
of the square root of x equals just plain old x?
- Why Is an Exponent of 1/2 the Same as a Square Root? [05/05/2004]
Why is raising a number to the 1/2 power the same as taking the square
root of the number? For example, 36^(1/2) = 6.