Ask Dr. Math

Middle School Archive

Browse Middle School Square 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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Negative Fractional Powers [12/08/1998]

How do you do problems like 3^(-2/3), where your power is a negative fraction?

Negative Square Roots [01/25/2003]

Why can't both 10 and -10 be the square roots of 100?

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?

Powers of Roots [05/14/2001]

What do you do when you have a small number in front of a square root? For example: 3(except it's a little three)sqrt of -216.

Principal or Positive Square Roots [08/31/2006]

Why do we only take the positive value for a square root when clearly something like the square root of 9 could be 3 or -3?

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?

Radical on the Bottom of a Fraction [12/10/2002]

Simplify: 4/3-root 5.

Rationalizing the Denominator [07/10/2003]

1 / ((sqrt)3 + (sqrt)5 + (sqrt)7)

Rational Numbers and Square Roots [10/07/1998]

Our teacher told us to find all the rational numbers we could containing a sqrt symbol...

Simplifying Fractions with Radicals [03/09/2006]

How do I simplify 18*sqrt(200) divided by 12*sqrt(500)?

Simplifying Square Roots [05/25/1999]

How do I simplify a square root?

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?

Square Root [03/15/1999]

Why does the square root of (9+4) not equal the square root of 9 plus the square root of 4?

Square Root Calculation [01/02/1998]

I can't recall how we used to calculate the square root of a number before we learned to push a button on the calculator.

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 Notation [11/14/2001]

Where do you write the answer to a square root?

Square Root of 100 [04/15/1999]

Why can't -10 be a square root of 100?

The Square Root of 2 [10/21/1995]

How many places do you actually know of the square root of number 2?

Square Root of a Negative Number Squared [01/23/1997]

Is Sqrt(-6)^2 equal to 6 or -6?

Square Roots [07/17/1998]

I am confused - what are square roots and what do they look like?

Square Roots [08/19/2003]

Why is a square root called a square root?

Square Roots and Calculator Rounding [01/12/1999]

When we find the square root of 43 on our calculators, we get different answers. Which is right?

Square Roots and False Solutions [5/22/1995]

Why does squaring a variable sometimes create a false solution?

Square Roots and Fractions [10/25/1998]

How do you simplify a square root of a fraction?

Square Roots and Irrational Numbers [10/08/1997]

Will every square root of a positive number that is not a perfect square be an irrational number?

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 impossible?

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 of Numbers between 0 and 1 [07/08/1999]

When taking the square root of a number repeatedly, why do square roots always come closer to the number one?

Square Roots Taken Repeatedly [11/21/2002]

When you repeatedly take the square root of a whole number, do you end up with 1?

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?

Table of Square Roots [05/12/2000]

Do you have a table of square roots that are already worked out?

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)]?

Using the Ones Digit to Estimate Square Roots [02/11/2005]

I am having trouble figuring out square roots quickly. For bigger numbers like 289, I have to try a lot of numbers before I get the answer. Is there some way to do this more easily?

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