Algebra 1
1.3: Square Roots
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KNOWLEDGE:
 The definition of a square root can be used to find the exact square roots of perfect squares. Nonperfect square numbers can be approximated with a calculator.
SKILLS:
 Identify perfect squares
 Evaluate and simplify square roots.
VOCABULARY:
 square root, radical, perfect square, inverse
Vocabulary
Definition of a square root:
A radical symbol
A perfect square
Perfect Square Chart
***Reading Math:
The expression √
does not represent a real number because there is no real number that can be
multiplied by itself to form a product of –36.
Example 1: Finding Square Roots of Perfect Squares
Directions: Find each square root.
A) √
B) √
The square roots of many numbers like√ are not whole numbers. A calculator can approximate the value of
√ as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the
square roots of other numbers.
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Algebra 1
Example 2: Approximating Square Roots
Directions: Use a calculator to approximate the square root. Round your answer to the nearest tenth.
A)√
B) √
Simplifying Square Roots
 To simplify a square root means to find another expression with the same value. It does NOT mean to
find a decimal approximation.
 To simplify a square root you must first find the largest perfect square which will divide evenly into the
number under your radical sign. This means that when you divide, you get no remainders, no decimals,
and no fractions.
 If the number under your radical cannot be divided evenly by any of the perfect squares, your radical is
already in simplest form and cannot be reduced any further.
Simplify: √
Step 1: First, find the largest perfect square which will divide evenly into the number under your radical sign.
This means that when you divide, you get no remainders, no decimals, and no fractions.
Step 2: Write the number appearing under your radical as the product of the perfect square and your answer
from dividing.
Step 3: Give each number in the product its own square root sign
Step 4: Reduce the perfect radical you have now created. You now have your answer.
**Caution
If instead of choosing 16 as the largest perfect square to start this process, you choose 4, look what happens……..
√
√
√ √
√
Unfortunately, this answer is not in simplest form. The 12 can also be divided by the perfect square of 4.
If you do not choose the largest perfect square to start the process, you will have to repeat the process.
Example 3: Simplifying Square Roots
Directions: Simplify each square root.
A)√
B) √
B) √
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Algebra 1
Example 4: Simplifying Square Roots
Directions: Simplify each square root.
Don’t let the number in front of the radical distract you. It is simply “along for the ride” and will be multiplied times our final
answer.
A) √
B) √
B) √
Example 5: Keystone Exam Prep
Directions: An expression is shown below.
√
√
Which value of x makes the expression equivalent to
A) 5
B) 25
C) 50
D) 100
?
Example 6: Keystone Exam Prep
Directions: An expression is shown below.
√
For which value of x should the expression be further simplified?
A) x = 10
B) x = 13
C) x = 21
D) x = 38
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