[MUSIC PLAYING] Let's look at our objectives for today. We'll start by defining square roots and perfect squares. We'll then look at the products property of roots. We'll do some examples simplifying square roots using the product property of roots. And finally, we'll do some examples using order of operations with square roots.
Let's start by looking at the definition of square roots and perfect squares. The square root of a number is an operation or calculation that is performed on a number. It is a number whose product with itself is the original number. For example, the square roof of 9 is 3, because 3 times 3 is 9.
We find square roots in many areas of art, design, and engineering. For example, the distance formula and Pythagorean theorem are significant geometric principles involving square roots. We also use square roots when solving quadratic equations, which can be used to model a ball or projectile under the influence of gravity.
We call 9, as well as other numbers, perfect squares because when we take the square root of the number, the answer is an integer. So we also call perfect squares squares of integers. Let's look at some other examples.
As you can see, 7 squared is 49, and the square root of 49 is 7. So 49 is a perfect square. 2 squared is 4. So the square root of 4 equals 2. So 4 is a perfect square. And 5 squared equals 25. So the square root of 25 equals 5. So 25 is also a perfect square. An example of a number which is not a perfect square is 20, because no integer squared equals 20, and because the square root of 20 is approximately 4.472, not at integer.
To simplify square roots, we can find factors of the number inside the square root and break the expression into several square roots using the product property of roots. In general, the product property of roots says that the square root of a times b is equivalent to the square root of a times the square root of b. For example, using 10 squared equals 100, we can see that the square root of 100 equals 10.
But we can also see this using the product property of roots. The square root of 100 is equal to the square root of 4 times 25, so using the product property of roots, we know this is equal to square root of 4 times the square root of 25. This is equal to 2 times 5, which equals 10.
The product property of roots also works for more than just two numbers, and is useful in simplifying square roots of numbers that are not perfect squares. However, be careful not to assume that this also works for addition, because in general, the square root of a plus b is not equal to the square root of a plus the square root of b. Here's an example of this.
The square root of 4 plus 9 is equal to the square root of 13, which is approximately 3.606. But the square root of 4 plus the square root of 9 is equal to 2 plus 3, which is equal to 5. So we can see that there is no addition property of roots.
So now let's see how we can simplify square roots that are not necessarily perfect squares. The product property is useful when simplifying the root of a number that is not a perfect square. However, recognizing the perfect squares will help us simplify. So here's another example of using the product property of roots.
We want to simplify the square roof of 50. 50 is not a perfect square, so we know that when we take the square root of 50, our answer will not be an integer. However, we want to write 50 as a product of one or more perfect squares to simplify. We know that 25 is a perfect square, so we can rewrite the square root of 50 as the square root of 25 times 2.
Using the product property of roots, this is the same as the square root of 25 times the square root of 2. Square root of 25 is 5, and so we now have 5 times the square root of 2. It is standard to put the square root at the end of the expression.
Here's another example. Suppose the baseball diamond in a stadium has an area of 8,100 square feet. Since all the sides of the diamond have equal length, the area of the diamond can be represented with the formula a is equal to s squared, where s is the side length. So if we know the area is 8,100, we have 8,100 equals s squared. Then to find the length of each side of the diamond, or the distance between each base, we can take the square root of each side of the equation, giving us the square root of 8,100 is equal to s.
Now we can simplify using the products property of roots. The square root of 8,100 is equal to the square root of 81 times 100. This is equal to the square root of 81 times the square root of 100, which equals 9 times 10, which is 90 feet. So the side length, or the distance between bases in the baseball diamond is approximately 90 feet.
Now let's look to see how we can simplify square roots using the order of operations. We have talked about the order of operations in the acronym PEMDAS, which is used to remember the order of operations. PEMDAS stands for Parentheses, and other grouping symbols, Exponents, Multiplication, Division, Addition, and Subtraction.
Square roots and other radicals fall under the parentheses, as a square root acts as a grouping symbol when there are other operations underneath the square root. This means that the operations underneath the radical must be performed before taking the root. Let's look at an example.
We want to simplify the square root of 3 times 12 plus 4 squared. Because the radical acts as a grouping symbol, everything underneath the radical must be evaluated before taking the root. So underneath the radical, order of operations still applies. So we start with our exponent. 4 squared is 16, so our expression is now the square root of 3 times 12 plus 16.
We then move on to multiplication. 3 times 12 is 36, so our expression is the square root of 36 plus 16. Next, we add 36 plus 16, which is 52, so our expression is the square root of 52. Now we can simplify the square root.
Square root of 52 can be written as the square root of 4 times 13, which is the square root of 4 times the square root of 13. We can simplify the square root of 4 to be 2, so our answer is 2 times the square root of 13. Again, we place the square root of 13 at the end of the expression.
Here's our last example. Suppose on a vacation you were traveling to a landmark that is 2 miles east and 4 miles south of your hotel. What is the straight line distance between your hotel and the landmark? We can use the Pythagorean theorem to determine this straight line distance.
The straight line distance, d, will be equal to the square root of 2 squared plus 4 squared. We start to simplify by evaluating our exponents and square 2 and 4. This gives us the square root of 4 plus 16. Square root of 4 plus 16 is the square root of 20, which can be written as the square root of 4 times 5.
This is equivalent to the square root of 4 times the square root of 5, which is equal to 2 times the square root of 5. Again, we would place the square root of 5 at the end of the expression. So the straight line distance between your hotel and the landmark is 2 square root of 5 miles.
Let's go over our important points from today. The square root of a number is an operation that is performed on a number. A perfect square is a number that is the square of a whole number. And to simplify the square roots, we can use the product property of roots.
So I hope that these key points and examples helped you understand a little bit more about simplifying square roots. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.
A number that is the square of a whole number.
A number whose product with itself is x.