MA.8.A.6.2 Make reasonable approximations of square roots & mathematical expressions that include square
roots, & use them to estimate solutions to problems & to compare mathematical expressions involving real
numbers & radical expressions.
DEFINITIONS
Square Roots: A square root of a number is one of its two equal factors. A radical sign,
square root. Every positive number has both a negative and positive square root.
, is used to indicate a positive
Real Number: The set of all rational and irrational numbers.
Rational Number: a real number that can be expressed as a ratio of 2 integers.
Irrational Number: a real number that cannot be expressed as a ratio of 2 integers
Radical Expression: has a root (square root/cube root).
Ex.
25 is a radical
Ex.
3
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EXAMPLES
Examples 1: Find each square root.
1)
Find the positive square root of 1; 12 = 1.
1
2)  16
3)  0.25
4)
 49
Find the negative square root of 16; (-4)2 = 16.
Find both square roots of 0.25; 0.52 = 0.25.
There is no real square root because no number times itself is equal to -49.
Example 2: Solve a2 =
a2 =
4
9
a2 = 
4
. Check your solution(s).
9
Write the equation.
4
9
2
2
a=
or 3
3
Definition of square root
Check
The equation has two solutions,
2 2 4
2
2 4
  and     .
3 3 9
3
3 9
2
2
and - .
3
3
1
Most numbers are not perfect squares. You can estimate square roots for these numbers.
Example 1 Estimate
204 to the nearest whole number.

The largest perfect square less than 204 is 196.

The smallest perfect square less than 204 is 225.
196 < 204 < 225
Write an inequality.
142 < 204 < 152
196 = 142 and 225 = 152.
Find the square root of each number.
Simplify.
So,
is between 14 and 15. Since 204 is closer to 196 than 225, the best whole number estimate for
Example 2 Estimate
to the nearest whole number.

The largest perfect square less than 79.3 is 64.

The smallest perfect square less than 79.3 is 81.
64 < 79.3 < 81
2
8 < 79.3 < 9
is 14.
Write an inequality.
2
64 = 82 and 81 = 92.
Find the square root of each number.
Simplify.
So,
is between 8 and 9. Since 79.3 is closer to 81 than 64, the best whole number estimate for
is 9.
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MA.8.A.6.2 Practice Problems
1.
3. The geometric mean of two numbers a and b can be
found by evaluating
. Estimate the geometric mean
of 5 and 10 to the nearest whole number.
2.
4. The radius r of a certain circle is given by
Estimate the radius of the circle to the nearest foot.
5. In a triangle whose base and height are equal, the base b
6. The radius of a cylinder with volume V and height 10
is given by the formula
, where A is the area of
the triangle. Estimate to the nearest whole number the base
of this triangle if the area is 17 square meters.
centimeters is approximately
. If a can that is 10
centimeters tall has a volume of 900 cubic centimeters,
estimate its radius.
7. The formula
can be used to find the speed s
of a car in miles per hour when the car needs d feet to come
to a complete stop after slamming on the brakes. If it took a
car 12 feet to come to a complete stop after slamming on the
brakes, estimate the speed of the car.
.
8. The formula for the area of a square is A = s2, where s is
the length of a side. Estimate the length of a side for each
square.
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9.
10. Andre will install carpet in his bedroom. The floor in
Andre’s bedroom is square, with an area of 65 square feet.
Which is closest to the length, in feet, of each side of
Andre’s bedroom floor?
A. 8
B. 9
C. 16
D. 17
11.
12.
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13.
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