Q&A for How to Add and Subtract Square Roots

Because (x - y)(x + y) = x² - y², the example you show is equal to 6 - 2, or 4.

√2 + √2 = 2√2 = √(2² x 2) = √8.

√2 + √2 = 2√2.

Here you're subtracting one square-root-of-8 from two square-roots-of-8. That leaves one square-root-of-8. √8 simplifies to 2√2.

2√48 - √3 = [2√(3 x 16)] - (√3) = [2√(3 x 4²)] - (√3) = [(2 x 4)√3] - (√3) = 8√3 - 1√3 = 7√3.

√4 = 2. Add 9.

(√3 + 2√3) ÷ 2 = (3√3) / 2.

Multiply 1.414 by two.

5√2 - 7√2 + 1√2 = -1√2 = -√2.

Change the fractions to equivalent fractions with the same denominator: 23/3 = 46/6, and 3/2 = 9/6. So 23/3 - 3/2 = 46/6 - 9/6 = 57/6.

Add 1.414 plus 1.414.

√3 = 1.732. 1.732 - 1 = 0.732.

2√3 - 1√3 + √12 = 1√3 + √(2² x 3) = 1√3 + 2√3 = 3√3.

It looks like some irrational number, but it turns out to be exactly 10. Here's how. Let x be the value of the expression, and square both sides so that x^2 = (45-√2000) + √ (45+√2000) + 2 √ (45-√2000) √ (45+√2000). Expand the cross term using difference of squares formula and simplify to x^2 = 90 + 2 √(45^2 - 2000) = 90 + 2√25 = 100. Since x is obviously the positive root of 100, x = 10.

2² + √2 = 4 + √2. You can leave the answer in that form, or you can express it as (4 + 1.414) = 5.414.

That equals (2)(√3) - 10)(√3) = -8√3. You can leave it in that form, or you can further evaluate: (-8)(1.732) = -13.856.

If you start with any number and add it to itself, you wind up with two of those numbers. In other words, x + x = 2x. Therefore, √2 + √2 = 2√2. If you want to evaluate that further, it's (2)(1.414) = 2.828.

First rationalize the denominator (get rid of the radical signs there). Do that by multiplying both numerator and denominator by (√5 - √3). The new numerator is (√5 - √3)², and the new denominator is 5 - 3 = 2. You can leave the answer in that form, or you can further evaluate: (2.24 - 1.73)² / 2 = (0.51)² / 2 = 0.26 / 2 = 0.13.

(√5)² + 5² = 5 + 25 = 30.

Multiply both numerator and denominator by √3. That's the same as multiplying by 1, which doesn't change the value of the expression. (12 / 7√3)(√3 / √3) = (12√3) / [( 7)(3)] = (12√3 / 21 = (4√3) / 7. There's now a square root (an irrational number) in the numerator but none in the denominator, which is what you want. (A rational denominator is considered easier to work with than an irrational one, although in practice it doesn't really make much difference, since you typically wind up with an irrational number in the numerator. Nonetheless, it is standard practice to rationalize denominators.)

(6√3) - 6 = (6)(1.732) - 6 = 10.392 - 6 = 4.392.

2√2. Any number added to itself is twice the original number.

First rationalize the denominators. Take (8 / √2) and multiply it by (√2 / √2), which is the same as multiplying it by 1. You get (8√2) / [(√2)(√2) = (8√2) / 2 = 4√2. Save that number for a moment. Now take (40 / √2) and multiply it by (√2 / √2). You get (40√2 / [(√2)(√2) = (40√2) / 2 = 20√2. Now multiply: (4√2)(20√2) = (80)(2) = 160.

This cannot be simplified, so you would have to use a calculator to find the exact value.

0. Anything subtracted from itself is zero. √2 - √2 = 0.

If you mean you are subtracting √6 from √6, anything subtracted from itself is zero. So √6 - √6 = 0.

-5xy² -(-4xy²) = -5xy² + 4xy² = -xy².

√(a² - b²) cannot be simplified. It can be rewritten as √[(a - b)(a + b)], but that's not really any simpler.

This cannot be reduced or simplified. It can be left in that form, or you can evaluate the square roots, then multiply and subtract. Thus, 3√6 - 3√2 = (3)(2.449) - (3)(1.414) = 7.347 - 4.242 = 3.105.

√8 + √8 = (2√2) + (2√2) = 4√2. You can leave the answer in that form, which is perfectly acceptable, or you can evaluate the radical: 4√2 = (4)(1.414) = 5.656.