Download Simplify Square Roots

Simplify Square Roots
When we discussed exponents, the square (exponent of two) of a
number means that we have two factors of the base. A square root
“unsquares” a number. In other words, we are asking “what number
squared gives us the base.”
The symbol that is used, √ , is called the radical sign, or in this case a
square root sign. The number or term underneath the radical sign is
called a radicand. Since the radicand represents a number squared, it
cannot be negative. However, the square root can be either positive or
negative: a positive term squared is positive, and a negative term
squared is also positive. A negative radicand indicates no real number.
The positive root, or answer, is called the principal root.
Example 1: Simplify the following: √49 , −√49 , and √−49
√49 = √7 · 7 = 7
The principal root
−√49 = −√7 ∙ 7 = - 7 The negative root
√−49 = Not a Real Number
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Not all radicands are perfect squares. When this occurs, the answer
would be a never-ending decimal number which would have to be
represented as a rounded approximation of the square root. To be as
accurate as possible, we use a property called the product rule of
radicals:
√𝒂 ∙ 𝒃 = √𝒂 ∙ √𝒃
The product rule is used to simplify the radicand into prime factors.
Then, since we are looking for a number squared, for each pair of
factors, one factor is written outside the radical sign. Numbers that are
not part of a pair are left as the radicand. If there is more than one factor
outside and/or inside the radical sign, those factors are multiplied.
Example 2: Simplify: √75
√75 = √5 ∙ 5 ∙ 3
= 5 √3
Factor the radicand
Example 3: Simplify: 5√63
5√63 = 5 ∙ √7 ∙ 3 ∙ 3
= 5 ∙ 3√7
= 15√7
Factor the radicand
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 4: Simplify: −√72
−√72 = −√3 ∙ 3 ∙ 2 ∙ 2 ∙ 2
= −( 3 ∙ 2)√2
= −6√2
Factor the radicand
Example 5: Simplify: √60𝑥 5 𝑦 3 𝑧 4
√60𝑥 5 𝑦 3 𝑧 4 = √5 ∙ 3 ∙ 2 ∙ 2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑧 ∙ 𝑧 ∙ 𝑧 ∙ 𝑧
= 2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑧 ∙ 𝑧√5 ∙ 3 ∙ 𝑥 ∙ 𝑦
= 2𝑥 2 𝑦 𝑧 2 √15𝑥𝑦
Notice that the above variables could have been simplified by dividing
the exponents by 2 (dividing by 2 tells us how many pairs of factors)
with the remainder left as part of the radicand.
√60𝑥 5 𝑦 3 𝑧 4 = √5 ∙ 3 ∙ 2 ∙ 2 ∙ 𝑥 4 ∙ 𝑥 ∙ 𝑦 2 ∙ 𝑦 ∙ 𝑧 4
= 2 ∙ 𝑥 4/2 ∙ 𝑦 2/2 ∙ 𝑧 4/2 √5 ∙ 3 ∙ 𝑥 ∙ 𝑦
= 2𝑥 2 𝑦 𝑧 2 √15𝑥𝑦
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)