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SAT Subject Test Mathematics Level 2: Tutoring Solution24 chapters | 231 lessons

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Lesson Transcript

Instructor:
*Sharon Linde*

How do you simplify an expression like the square root of 50? It's not a perfect square, but it does factor. This lesson goes over the rules and process for simplifying numbers underneath the square root symbol.

You have just graduated from high school and are working a summer job as a handyman before going to college in the fall. Your first assignment is to build fencing for a small vegetable garden. The area enclosed should be exactly 50 square feet, and the fencing material you have will only allow you to build an enclosure that has right angles. You are happy to hear that you'll receive a bonus if you can use the least amount of fencing possible, but how do you do that?

You recall that the largest area you can make with a limited perimeter using right angles is a square. You also know that the length of each side of a square will be the square root of the area. Since you know the total area of the garden will have to be 50 square feet, now all you have to do is find the square root of 50.

**Step 1:** Write the square root of 50

This step is so simple, we almost left it out! But we need to start somewhere, and writing 50 underneath the square root symbol is an excellent place to start. This symbol is what is used in mathematics to let anyone viewing it know that what's under this symbol is different than anything outside of the square root symbol.

**Step 2:** Factor 50 under the square root symbol

The first step in simplifying the expression of the square root of 50 involves finding **factors** of 50. This just means we are trying to find two whole numbers that, when multiplied, equal 50. Since 50 is an even number, 2 is going to be a factor, and so we can rewrite 50 as 2 times 25. Of course, since 50 was underneath the square root symbol, then its factors also have to be under that same symbol.

**Step 3:** Continue factoring until you have all prime numbers

To simplify a number underneath the square root symbol, it is very useful to keep factoring the factors until the only factors that are left are prime numbers. In this problem that is just one more step, because the only way to factor 25 is to break it up into 5 times 5. When trying to simplify the square root of other numbers, you may have to continue factoring for multiple steps beyond this one.

**Step 4:** Combine factors using exponents

Repeated factors can be rewritten more efficiently by using exponents. In our problem, we can combine the two factors equal to 5 into one factor that is equal to a base of 5 with an exponent of 2. Another way to say a base of 5 with an exponent of 2 is just 5 squared.

**Step 5:** Move bases outside of square root symbol, if possible

When using the square root symbol, the rule for moving bases outside of the symbol is to divide the exponent by 2. Of course, you only want to do this for even exponents; otherwise they won't divide evenly, and you'd be left with a fractional exponent. Although this is useful in some circumstances, it really wouldn't help us simplify our current problem.

We have two bases underneath the square root symbol: 2 and 5. The 2 only has an exponent of 1, so we leave that where it is. 5 has an exponent of 2, which is evenly divisible by 2, so we can move it outside the square root symbol using our rule. When base of 5 moves outside the square root symbol, the exponent of 2 gets divided by 2. Since the exponent was 2, it becomes 1 on the outside of the square root symbol. So, 5 raised to the first power is just 5.

Our last step gives us our solution: 5 times the square root of 2 is the simplified version of the square root of 50. To go any further, we would need to use something to turn the square root of 2 into a decimal number, such as a calculator, table of square roots, or memorization. In general, remember that when moving from underneath the square root symbol to outside of it, the base stays the same and the exponent is divided by 2.

But, that solution doesn't help you with your current handyman fencing problem! Luckily, you remember from one of your math classes that the square root of 2 is equal to 1.41. Multiplying that by 5 gives you an answer of 7.05.

Simplifying the square root of 50 seems pretty straight forward, but it's a little more complicated than you might expect. In general, you have to remember that when moving from underneath the square root symbol to outside of it, the base stays the same and the exponent is divided by 2. However, the step-by-step process includes five steps that include:

**Step 1:** Writing down the square root itself, which is in this case, the square root of 50

**Step 2:** Factor 50 under the square root symbol

**Step 3:** Continue factoring until you have all prime numbers

**Step 4:** Combine factors using exponents

**Step 5:** Move bases outside of square root symbol, if possible

Your ultimate solution should be 5 times the square root of 2 is the simplified version of the square root of 50, which, when simplified, should be 7.05. Keep in mind that these steps can apply to any square root calculation. So, maybe it wasn't all that complicated after all!

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SAT Subject Test Mathematics Level 2: Tutoring Solution24 chapters | 231 lessons

- Evaluating Square Roots of Perfect Squares 5:12
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- 2 Times the Square Root of 2: How-to & Steps 5:37
- Simplifying the Square Root of 50 5:11
- Go to Square Roots: Tutoring Solution

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