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Study Guides > College Algebra

Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since 42=16{4}^{2}=16, the square root of 1616 is 44. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root. In general terms, if aa is a positive real number, then the square root of aa is a number that, when multiplied by itself, gives aa. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals aa. The square root obtained using a calculator is the principal square root. The principal square root of aa is written as a\sqrt{a}. The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression. The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.

A General Note: Principal Square Root

The principal square root of aa is the nonnegative number that, when multiplied by itself, equals aa. It is written as a radical expression, with a symbol called a radical over the term called the radicand: a\sqrt{a}.

Q & A

Does 25=±5\sqrt{25}=\pm 5?

No. Although both 52{5}^{2} and (5)2{\left(-5\right)}^{2} are 2525, the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25=5\sqrt{25}=5.

Example 1: Evaluating Square Roots

Evaluate each expression.
  1. 100\sqrt{100}
  2. 16\sqrt{\sqrt{16}}
  3. 25+144\sqrt{25+144}
  4. 4981\sqrt{49}-\sqrt{81}\\

Solution

  1. 100=10\sqrt{100}=10 because 102=100{10}^{2}=100
  2. 16=4=2\sqrt{\sqrt{16}}=\sqrt{4}=2 because 42=16{4}^{2}=16 and 22=4{2}^{2}=4
  3. 25+144=169=13\sqrt{25+144}=\sqrt{169}=13 because 132=169{13}^{2}=169
  4. 4981=79=2\sqrt{49}-\sqrt{81}=7 - 9=-2 because 72=49{7}^{2}=49 and 92=81{9}^{2}=81

Q & A

For 25+144\sqrt{25+144}, can we find the square roots before adding?

No. 25+144=5+12=17\sqrt{25}+\sqrt{144}=5+12=17. This is not equivalent to 25+144=13\sqrt{25+144}=13. The order of operations requires us to add the terms in the radicand before finding the square root.

Try It 1

Evaluate each expression.

a. 225\sqrt{225} b. 81\sqrt{\sqrt{81}} c. 259\sqrt{25 - 9} d. 36+121\sqrt{36}+\sqrt{121}

Solution

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  • College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface. License: CC BY: Attribution.