Example Expressions: sqrt(32s), 2sqrt(8s), 8ssqrt(2s) What They Have in Common: s and a factor of 2 under the radical
Practice makes perfect
We are asked to write three different radical expressions that have 4sqrt(2s) as their simplified form. In order to do this, let's rewrite the given expression in three different ways using the Properties of Square Roots.
First Expression
First, we can rewrite the given expression by moving 4 under the square root. To do this, we need to determine a square root of what number gives us 4.
sqrt(16)=4
Now that we know this, we can replace 4 with sqrt(16). Then, using the Multiplication Property of Square Roots, we can rewrite the product as one square root.
We can conclude that sqrt(32s) simplifies to 4sqrt(2s).
Second Expression
We can form the second expression in a similar way. However, in this case, we will not rewrite 4, but one of its factors, 2. We can use the fact that 2 is equal to sqrt(4).
4= 2* 2 = 2* sqrt(4)
Next, we will apply the Multiplication Property of Square Roots again to combine the two roots.
Therefore, another expression that can be simplified to 4sqrt(2s) is 2sqrt(8s).
Third Expression
Finally, let's form one more expression. Note that if s is not equal to 0, the radicand 2s can be rewritten as 4s^22s.
2s*2s/2s=4s^2/2s
Then, we can apply the Division Property of Square Roots.
A third expression with a simplified form of 4sqrt(2s) is 8ssqrt(2s).
Comparison
Let's now compare all three of our newly created expressions.
sqrt(32s), 2sqrt(8s), 8s/sqrt(2s)
What do they have in common? Each of these expressions has a square root and they all have s and a factor of 2 under the radical.
sqrt(16* 2s), 2sqrt(4* 2s), 8s/sqrt(2s)
