Square roots and operations
Pre-algebra students in OpenStax Prealgebra 2E learn that the radical sign acts as a grouping symbol: simplify everything inside before performing other operations. The square root of (9 + 16) requires adding first to get the square root of 25 = 5, not adding sqrt(9) + sqrt(16) = 3 + 4 = 7. These two expressions have different values. Additionally, no real number square root exists for a negative number, because squaring any real number always produces a non-negative result.
Key Concepts
Property When using the order of operations, treat the radical sign as a grouping symbol. Simplify any expressions under the radical sign before performing other operations. For any negative number, there is no real number solution for its square root.
Examples To simplify $\sqrt{9 + 16}$, you must first add the numbers inside the radical to get $\sqrt{25}$, which equals $5$. To simplify $\sqrt{9} + \sqrt{16}$, you find each square root separately before adding: $3 + 4 = 7$. Note this is different from the first example. The expression $\sqrt{ 169}$ is not a real number because no real number multiplied by itself can result in $ 169$.
Explanation The radical sign acts like parentheses, so you must simplify the expression inside it first. You cannot take the square root of a negative number in the real number system because squaring any real number always results in a non negative value.
Common Questions
How does order of operations apply to square roots?
Treat the radical sign like parentheses. Simplify the entire expression inside the radical before taking the square root.
What is the square root of (9 + 16)?
First add: 9 + 16 = 25. Then take the square root: sqrt(25) = 5.
Is sqrt(9) + sqrt(16) the same as sqrt(9 + 16)?
No. sqrt(9) + sqrt(16) = 3 + 4 = 7, while sqrt(9 + 16) = sqrt(25) = 5. These are different values.
Why is there no real square root of a negative number?
Squaring any real number (positive or negative) always gives a non-negative result, so no real number squared can equal a negative number.
What symbol is used for square roots and how is it read?
The radical sign is the symbol. The expression sqrt(25) is read as 'the square root of 25' and equals 5 because 5 squared is 25.