Square Roots Study Guide (page 2)
Introduction to Square Roots
Math is radical!
This lesson exposes the meaning behind the sneaky √ symbol. What are roots and radicals? How can you estimate them?
Taking the square root (also called a radical) is the way to undo the exponent from an equation like 22 = 4. If 22 = 4, then 2 is the square root of 4 and √4 = 2. The exponent in 22 tells you to square 2. You multiply 2 · 2 and get 22 = 4.
The radical sign √ indicates that you are to find the square root of the number beneath it. The number inside the radical sign is called the radicand. For example, in √9, the radicand is 9.
A positive square root of a number is called the principal square root. For √9, 3 × 3 = 9, so 3 is the principal square root.
A negative sign outside the radical sign means the negative square root. For example, √9 = 3, but –√9 = –3.
The easiest radicands to deal with are perfect squares. A perfect square is a number with an integer for a square root. For example, 25 is a perfect square because its square root is 5, but 24 is not a perfect square because it doesn't have an integer for a square root. (No number multiplied by itself equals 24.)
Because they appear so often, it is useful to learn to recognize the first few perfect squares:
- 02 = 0
- 12 = 1
- 22 = 4
- 32 = 9
- 42 = 16
- 52 = 25
- 62 = 36
- 72 = 49
- 82 = 64
- 92 = 81
- 102 = 100
- 112 = 121
- 122 = 144
It is even easier to recognize when a variable is a perfect square because the exponent is even. For example, x 14 = x 7 · x 7 and a 8 = a 4 · a 4.
You can use known perfect squares to estimate other square roots. Suppose you were asked to determine √50. What is the closest perfect square to 50? Try a few numbers to see:
So, you can determine that √50 is going to be between 7 and 8, much closer to 7.
To determine if a radicand contains any factors that are perfect squares, factor the radicand completely. All the factors must be prime. A number is prime if its only factors are 1 and the number itself. A prime number cannot be factored any further. Let's try determining the square root of the following:
√ 64x 2y10
Write the number under the radical sign as a square:
√8xy5 · 8xy5
Because you have two identical terms multiplied by each other, you know this is a perfect square. Evaluate to find the final solution: 8xy5. You could also have split the radical into parts and evaluated them separately.
Let's try another one. Find the square root of √ 64 x2 y 10.
First, split the terms:
√ 64 · x2 · y 10
Each term is a perfect square. Write as squares:
√8 · 8 · √x · x · √y5 · y5
Finally, evaluate each new radical:
8 · x· y5
Not all radicands are perfect squares. There is no whole number that, when multiplied by itself, equals 5. With a calculator, you can get a decimal that squares very close to 5, but it won't come out exactly. The only precise way to represent the square root of five is to write √5. It cannot be simplified any further.
There are three rules for knowing when a radical cannot be simplified any further:
- The radicand contains no factor, other than 1, that is a perfect square.
- The radicand cannot be a fraction.
- The radical cannot be in the denominator of a fraction.
Adding and Subtracting
Square roots are easy to add or subtract. You can add or subtract radicals if the radicands are the same. To add or subtract radicals, add the number in front of the radicals and leave the radicand the same. When you add 15√2 and 5√2, you add the 15 and the 5, but the radicand √2 stays the same. The answer is 20√2.
Multiplying and Dividing Radicals
To multiply radicals like 4√3 and 2√2, multiply the numbers in front of the radicals: 4 times 2. Then, multiply the radicands: 3 times 2. The answer is 8√6
- Try the following:
- 5√3 · 2√2
Multiply the numbers in front of the radicals. Then, multiply the radicands. You will end up with 10√6.
To divide the radical 4√6 by 2√3, divide the numbers in front of the radicals. Then, divide the radicands. The answer is 2√2.
As opposed to adding or subtracting, the radicands do not have to be the same when you multiply or divide radicals.
Find practice problems and solutions for these concepts at Square Roots Practice Questions.
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