Radicals with the same value may be written in different forms. Consider the following calculation:
\begin{align*}
\left(2\sqrt{2}\right)^2 \amp = 2^2\left(\sqrt{2}\right)^2~~~~~~~~\blert{\text{By the fourth law of exponents.}}\\
\amp = 4(2)=8
\end{align*}
This calculation shows that \(2\sqrt{2}\) is equal to \(\sqrt{8}\text{,}\) because the square of \(2\sqrt{2}\) is \(8\text{.}\) You can verify on your calculator that \(2\sqrt{2}\) and \(\sqrt{8}\) have the same decimal approximation, \(2.828\text{.}\)
Example9.37.
Show that \(~~\sqrt{45} = 3\sqrt{5}\text{.}\)
Solution.
\(\sqrt{45}\) is a number whose square is \(45\text{.}\) We can square \(3\sqrt{5}\) as follows:
Because the square of \(3\sqrt{5}\) is equal to \(45\text{,}\) it is the case that \(~3\sqrt{5}=\sqrt{45}\text{.}\)
It is usually helpful to write a radical expression as simply as possible. The expression \(2\sqrt{2}\) is considered simpler than \(\sqrt{8}\text{,}\) because the radicand is a smaller number. Similarly, \(3\sqrt{5}\) is simpler than \(\sqrt{45}\text{.}\) In this section we discover properties of radicals that help us simplify radical expressions.
In the Activities we will verify the following properties of radicals.
\(\sqrt{4} + \sqrt{9} = 2+3 = 5\text{,}\) but \(~\sqrt{13} \not= 5\text{,}\) so the first statement is false.
\(\sqrt{4}\sqrt{9} = 2(3) = 6\text{,}\) and \(~\sqrt{36} = 6\text{,}\) so the second statement is true.
SubsectionSimplifying Square Roots
We can use the product rule for radicals to write \(~\sqrt{12} = \sqrt{4}\sqrt{3}\text{.}\) Now, \(\sqrt{4}=2\text{,}\) so we can simplify \(\sqrt{12}\) as
The expression \(2\sqrt{3}\) is a simplified form for \(\sqrt{12}\text{.}\) The factor of \(4\text{,}\) which is a perfect square, has been removed from the radical. This example illustrates a strategy for simplifying radicals.
To Simplify a Square Root.
Factor any perfect squares from the radicand.
Use the product rule to write the radical as a product of two square roots.
Simplify the square root of the perfect square.
Example9.40.
Simplify \(~\sqrt{45}\text{.}\)
Solution.
We look for a perfect square that divides evenly into \(45\text{.}\) The largest perfect square that divides \(45\) is \(9\text{,}\) so we factor \(45\) as \(9 \cdot 5\) Then we use the product rule to write
but \(6.708\) is not the exact value for \(\sqrt{45}\text{.}\) For long calculations, too much accuracy may be lost by approximating each radical.
However, \(3\sqrt{5}\) is equivalent to \(\sqrt{45}\text{,}\) which means that their values are exactly the same. We can replace one expression by the other without losing accuracy.
QuickCheck9.42.
What is the difference between simplifying a square root and approximating a square root?
Answer.
Simplifying a square root gives an equivalent expression.
SubsectionSquare Root of a Variable Expression
To square a power of a variable we double the exponent. For example, the square of \(x^5\) is
Because taking the square root of a number is the opposite of squaring a number, to take the square root of an even power we divide the exponent by \(2\text{.}\)
How can we simplify the square root of an odd power? We write the power as a product of two factors, one having an even exponent and one having exponent 1.
Example9.46.
Simplify \(~~\sqrt{x^7}\)
Solution.
We factor \(x^7\) as \(~x^6 \cdot x\text{.}\) Then we use the product rule to write
How do we simplify the square root of a variable raised to an odd exponent?
Answer.
Write the power as a product of two factors, one having an even exponent and one having exponent 1.
If the radicand contains more than one variable or a coefficient, we consider the constants and each variable separately. We try to remove the largest factors possible from the radicand.
Example9.48.
Simplify \(~~\sqrt{20x^2y^3}\)
Solution.
We look for the largest perfect square that divides 20; it is 4. We write the radicand as the product of two factors, one of which contains the perfect square and even powers of the variables. That is,
\begin{equation*}
20x^2y^3 = 4x^2y^2 \cdot 5y~~~~~\blert{\text{Factor into perfect squares and "leftovers."}}
\end{equation*}
So, we cannot simplify a sum or difference if the expressions under the radical are different. However, we can combine radicals with the same radicand. For example, we can write
Square roots with identical radicands are called like radicals.
Look Closer.
We can add or subtract like radicals in the same way that we add or subtract like terms, namely by adding or subtracting their coefficients. For example,
\begin{equation*}
2r+3r=5r
\end{equation*}
where \(r\) is a variable that can stand for any real number. In particular, if \(r=\sqrt{2}\text{,}\) we have
However, \(~3\sqrt{2}~\) and \(~4\sqrt{3}~\) are not like radicals. We cannot simplify sums or differences of unlike radicals. Thus, \(~3\sqrt{2}+4\sqrt{3}~\) cannot be combined into a single term.
QuickCheck9.52.
How do we add or subtract like radicals?
Answer.
By adding or subtracting their coefficients
QuickCheck9.53.
How do we add or subtract unlike radicals?
Answer.
Unlike radicals cannot be combined.
Sometimes we must simplify the square roots in a sum or difference before we can recognize like radicals.
