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More Simplification / Multiplication (page 2 of 7)

Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains


Variables in a radical's argument are simplified in the same way: whatever you've got a pair of can be taken "out front".

  • Simplify sqrt[16 x^4]

    sqrt[16 x^4] = sqrt[(4 * 4) * (x * x) * (x * x)] = 4 * x * x = 4x^2

  • Simplify sqrt[12 a^4 b^7 c^3]
  • The 12 is the product of 3 and 4, so I have a pair of 2's but a 3 left over. Also, I have two pairs of a's; three pairs of b's, with one b left over; and one pair of c's, with one c left over. So the root simplifies as:

      sqrt[12 a^4 b^7 c^3] = sqrt[3 * 4 * aa * aa * bbb * bbb * b * c * c * c] = 2 * aa * bbb * c * sqrt[3 * b * c] = 2 a^2 b^3 c sqrt[3bc]

You are used to putting the numbers first in an algebraic expression, followed by any variables. But for radical expressions, any variables outside the radical should go in front of the radical, as shown above.

  • Simplify sqrt[20 r^18 s t^21]
  • Writing out the complete factorization would be a bore, so I'll just use what I know about powers. The 20 factors as 4×5, with the 4 being a perfect square. The r18 has nine pairs of r's; the s is unpaired; and the t21 has ten pairs of t's, with one t left over. Then:

      sqrt[20 r^18 s t^21] = sqrt[4 * r^18 * t^20 * 5 * s * t] = 2 * r^9 * t^10 * sqrt[5 * s * t] = 2 r^9 t^10 sqrt[5st]


 

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Technical point: Your textbook may tell you to "assume all variables are positive" when you simplify. Why? The square root of the square of a negative number is not the original number. For instance, you could start with –2, square to get +4, and then take the square root (which is defined to be the positive root) to get +2. You plugged in a negative and ended up with a positive. Sound familiar? It should: it's how the absolute value works: |–2| = +2. Taking the square root of the square is in fact the technical definition of the absolute value. But this technicality can cause difficulties if you're working with values of unknown sign; that is, with variables. The |–2| is +2, but what is the sign on | x |? You can't know, because you don't know the sign of x itself — unless they specify that you should "assume all variables are positive", or at least non-negative (which means "positive or zero").


Multiplying Square Roots

The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots.

Simplifying multiplied radicals is pretty simple. We use the fact that the product of two radicals is the same as the radical of the product, and vice versa.

  • Write as the product of two radicals:  sqrt(6)
    • sqrt(6) = sqrt(2 * 3) = sqrt(2) * sqrt(3)   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

Okay, so that manipulation wasn't very useful. But working in the other direction can be helpful:

  • Simplify by writing with no more than one radical: sqrt[2] sqrt[8]
    • sqrt[2] sqrt[8] = sqrt[2 * 8] = sqrt[16] = sqrt[4 * 4] = 4

  • Simplify by writing with no more than one radical: sqrt[3] sqrt[6]
    • sqrt[3] sqrt[6] = sqrt[3 * 6] = sqrt[3 * 3 * 2] = 3 sqrt[2]

  • Simplify by writing with no more than one radical:  sqrt(6) * sqrt(15) * sqrt(10)
    • sqrt(6) * sqrt(15) * sqrt(10) = sqrt (2 * 2 * 3 * 3 * 5 * 5) = 2 * 3 * 5 = 30

The process works the same way when variables are included:

  • Simplify by writing with no more than one radical: sqrt[4x] sqrt[5 x^3]
    • sqrt[4x] sqrt[5 x^3] = sqrt[4 * 5 * x * x^3] = sqrt[(2 * 2) * 5 * (x^2 * x^2)] = 2x^2 sqrt[5]

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Cite this article as:

Stapel, Elizabeth. "More Simplification / Multiplication." Purplemath. Available from
    http://www.purplemath.com/modules/radicals2.htm. Accessed
 

 

 

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