Square root : 10th grade math worksheets with answers in PDF.

10th grade math worksheets

Math worksheets on square roots in 11th grade of high school in order to assimilate all the properties of the square root and its definition. This list of sheets is accompanied by detailed answers to practice and review online in preparation for a test. You can also download in PDF or print these free documents for the tenth grade (10th).

Exercise 1:

We pose $\,E=(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})-8\sqrt{5}(\sqrt{5}-1)$ .

Write E in the form $\,a+b\sqrt{5}$ .

( a and b being relative numbers) .

Exercise 2:

Calculate D and E and give the results in the form $\,a\sqrt{b}$ where a and b are integers with b as small as possible.

$\,D=2\sqrt{12}-5\sqrt{27}+7\sqrt{75}$

$\,E=(\sqrt{2}+\sqrt{3})^2-5$

Exercise 3:

We give:

$\,A=\sqrt{12}+5\sqrt{75}-2\sqrt{27}$

$\,B=(5+\sqrt{3})^2-(2\sqrt{7})^2$

Write A in the form $\,a\sqrt{3}$ and B in the form $\,b\sqrt{3}$ where a and b are two relative integers .

Exercise 4:

We pose:
$\,a=\sqrt{3}(1+\sqrt{6})\,;\,\,b=3-\sqrt{6}$

1. Calculate a², b² and a²+b² .

2. Show that a²+b² is an integer.

3. If a and b are the lengths of the sides of the right angle in a triangle, what is the length of the hypotenuse?

Exercise 5: a rectangular room.
A rectangular room whose length is double the width has an area of 12.5 m².
What are its dimensions?

Exercise 6: square root and gcd.

1. Without calculating their PGCD, explain why the numbers 648 and 972 are not prime to each other.

2. a. Calculate PGCD ( 972 ; 648 ) by explaining the method used.

b. Show that $\sqrt{648}+\sqrt{972}=18(\sqrt{3}+\sqrt{2})$ .

Exercise 7: Pythagorean theorem.

Consider the following figure, we have (KH) // (AB).

a. calculate the exact values of AC and AB.

b. Prove that triangle ABC is right-angled at A.

c. Calculate the exact value of KH.

Exercise 8

Mental calculation
$\,\sqrt{1}\,,\,\sqrt{0,04}\,,\,\sqrt{64}\,,\,\sqrt{10000}\,,\,-\sqrt{36}$ .

Exercise 9

One student wrote:
$\,\sqrt{14}=7\,,\,\sqrt{9}=3\,$

Is he right, (to be justified).

Exercise 10
Are the following numbers equal to 3 or – 3?

$\,{\sqrt{3}}^2\,,\,-\sqrt{9}\,,\,(-\sqrt{3})^2\,,\,-\sqrt{3^2}\,,\,\sqrt{(-3)^2}\,,\,-{\sqrt{3}}^2$ .

Exercise 11
a. A square has an area of 13 cm². How big is its side ?

b. A square has side $\,\sqrt{6}$ cm .

What is its area?

Exercise 12
Simplify:

$\,\sqrt{4}\times \sqrt{9}\,,\,\sqrt{0,01}\times \sqrt{225}\,,\,\sqrt{2^2\times \,3^2\times \,5^2}\,,\,\sqrt{\frac{4}{9}}\,,\,\sqrt{\frac{100}{81}}\,,\,\sqrt{\frac{30}{7}}\times \,\frac{\sqrt{21}}{\sqrt{40}}$ .

Exercise 13
a. Write in the form $\,a\sqrt{b}$ the following expressions:

$\,\sqrt{8}\,,\,\sqrt{54}\,,\,\sqrt{500}\,,\,\sqrt{0,07}\,,\,-\sqrt{125}$ .
b. Write in the form $\,a\sqrt{7}$ ( a integer) :

$\,A=\,\sqrt{63}\,+\,2\sqrt{28}\,-\,\sqrt{700}$
$\,B=\,2\sqrt{28}\,-\,\sqrt{175}$

Exercise 14
Develop and reduce :
$\,A=\,(\sqrt{5}\,+\,\sqrt{7})^2$
$\,B=\,(\sqrt{7}\,-\,1)^2$
$\,C=\,(2\sqrt{2}-3)(2\sqrt{2}+3)$

Exercise 15

Show that E = 0.
$\,E=\,3\sqrt{54}-7\sqrt{6}-\sqrt{2}\times \,\sqrt{12}$

Exercise 16: golden rectangle.
A rectangle is called a golden rectangle when the quotient of its length and width is equal to the golden ratio.

1. Construct an ADEF square of side 6 cm.

Place the middle I of [DE].

Draw an arc of a circle with center I, radius IF, as in the figure below.

Complete the construction as below.

2. show that ABCD is a golden rectangle.

Exercise 17: simplify square roots.
We give: $A=\sqrt{300}-4\sqrt{27}+6\sqrt{3}\\B=(5+\sqrt{3})^2\\C=(3\sqrt{2}+\sqrt{5})(3\sqrt{2}-\sqrt{5})$

a) Write A in the form $a\sqrt{3}$ , where a is an integer.
b) Write B in the form $e+f\sqrt{3}$ , with e and f integers.

c) Show that C is an integer.

Exercise 18: complex equality
Prove, without using the calculator, that :

$\frac{\sqrt{11}}{2\sqrt{3}-\sqrt{11}}=2\sqrt{33}+11$

Exercise 19: development and complex expression
Develop and give the result as

as simplified as possible.

$K=2\sqrt{5}\times \,(3\sqrt{5}-1)-(3\sqrt{5}+2)\times \,(3\sqrt{5}-2)$

Exercise 20: expand and reduce this expression.
A = (27 -9)(27 )+9

Exercise 21: Aligned points.
Let there be three points O, U and I such that: $UI=\sqrt{63}$ ; $OU=\sqrt{343}$ and $OI=\sqrt{700}$ .

Are the points O, U and I aligned? Justify.

Exercise 22: Expand and reduce roots.
Expand and reduce the following expressions and give the result in the form $a+b\sqrt{c}$ ,

where a and b are relative integers and c is a positive integer.

$D=\sqrt{5}(\sqrt{5}-1)$

$E=\sqrt{2}(\sqrt{2}-5)-7\sqrt{2}$

Exercise 23: geometry.
Express the areas of these three rectangles as $a+b\sqrt{5}$

(where and are integers).

Exercise 24: Cheese bell and half circle.
we have a cheese bell which is a half-sphere of radius 9 cm. What is the maximum height of a cheese in the shape of a cylinder with a radius of 7 cm that can fit under this bell?
Explain.

Exercise 25: study of a cube.
ABCDEFGH is a cube of 4 cm edge.

a. Calculate the exact value of GD and write the result in the form $a\sqrt{2}$ with an integer.

b. What is the perimeter of the triangle BDG ?

Give the result in the form $a\sqrt{2}$ with an integer.

c. Calculate the exact value of GK.

d. Calculate the area of the triangle BGD.

Give the exact value and then a value rounded to the hundredth.

Exercise 26: Theodore of Cyrene’s spiral.
Observe the figure below.

a. Knowing that the triangle ABC is a right triangle isosceles in A, calculate the exact value of BC.

b. Using the figure below and question a, calculate the exact values of DB and EB.

Exercise 27: diagonals of a square.
Consider an ANIM square.

a. Calculate the exact length of the diagonal AI of the square MANI.

b. If $AN=a\,(a>0)$ , what is the length AI ?

Exercise 28: areas of triangles.
Using the data in the figure, determine the area of triangle ABC.

The proportions are not respected.

Exercise 29: develop a product.
Develop and reduce: $(\sqrt{2}+3)(4-5\sqrt{2})$ .

Exercise 30: Expand using the remarkable identities.
$A=(1+\sqrt{2})^2$

$B=(2\sqrt{3}+4)^2$

$C=(\sqrt{5}-\sqrt{6})(\sqrt{5}+\sqrt{6})$

$D=(\sqrt{7}-\sqrt{2})^2$

Exercise 31: Simplification.
Simplify the following expressions into the form $a\sqrt{b}$ , where a is a relative integer and b is the smallest possible integer.

$A=8\sqrt{7}-2\sqrt{28}+\sqrt{112}$

$B=2\sqrt{24}-3\sqrt{96}+9\sqrt{294}$

Exercise 32: Pythagorean snail.

Using the diagram opposite, draw a segment

of length $\sqrt{8}$ cm.

Explain your reasoning.

Exercise 33: Numerical functions
Consider the function h such that

1. Compute the image of 1 by the function h.

2. Compute the image of $\sqrt{2}$ by the function h.

3. Compute the image of $2\sqrt{3}$ by the function h.

Exercise 34: Volumes in space.
A glass of the conical form has a height of 11 cm.

What must be the exact value of the length of its diameter, in cm, for it to

contains 25 cL ?

Exercise 35: Diagonals of a flying deer.

Exercise 36: study of a square.
We consider the following figure. The unit is the centimeter.

a. Write $5\sqrt{12}-\sqrt{75}$ in the form $a\sqrt{b}$ , where and are relative integers, being the smallest possible.

b. What is the exact nature of ABCD? Justify your answer.

c. determine the perimeter of ABCD in the simplest form possible.

Then, give the rounding to the millimeter.

d. Determine the exact value of the area of ABCD.

Exercise 37: simplification of square roots.
Write in the form $a\sqrt{3}$ , a being a natural number:

$A=\sqrt{27}+7\sqrt{75}-\sqrt{300}$ .

Write in the form $p+m\sqrt{3}$ where m and p are relative integers :

$A=(3\sqrt{3}-2)(4-\sqrt{3})$

Exercise 38: square roots.
Put the following numbers in the form $a\sqrt{b}$

where and are two integers and the smallest possible.

$t=\sqrt{96}\,\,\,;\,\,\,u=\sqrt{108}\,\,\,;\,\,\,v=\sqrt{162}$

Exercise 39: calculation with roots.
Put the following numbers in the form $a\sqrt{5}$ .

$x=\sqrt{125}\,\,\,;\,\,\,y=\sqrt{500}\,\,\,;\,\,\,z=\sqrt{80}$

Exercise 40: volume of a prism.
Calculate the volume of a right prism of height $\sqrt{20}$ cm and base a triangle of dimensions: 1 cm; 3cm and $\sqrt{10}$ cm.

Write the result in the form $a\sqrt{b}$ ( where and are integers ) .

Exercise 41: Thales’ theorem.
The lines (AB) and (ED) are parallel.

$BC=\sqrt{5}$ ; $CD=\sqrt{3}$ and .

Lengths are in centimeters.

Calculate the length of the segment [AC].

Exercise 42: equations of the type x²=a.
Solve the following equations ( justify your answers ) :

a) 3x² =75

b) x²= -36

c) 25x² =4

d) 49x² = -64 e) x²+9=0

f) 27x² = 12

Exercise 43: fractions and product of square roots.
Calculate the following product and give the result in the simplest form possible.

$A=\frac{3\sqrt{5}-1}{\sqrt{5}+2}\times \,\frac{-\sqrt{15}}{\sqrt{5}-2}$

Exercise 44: square root problem.
Let a= $\sqrt{5}$ (1- $\sqrt{2}$ ) and b=5+ $\sqrt{2}$ .

a. Calculate a² and b².

b. Deduce the values of a²+b² and $https://mathovore.fr/cgi-bin/mimetex.cgi?\sqrt{a²+b²}$ .

Exercise 45: square root and rectangle.
ABCD is a rectangle such that :

$AB=(\sqrt{27}+\sqrt{3})\,\,cm$ and $BC=\sqrt{48}\,\,cm$ .

a) Prove that ABCD is a square.

b) calculate its perimeter and area.

Exercise 46:

Simplify the following expressions into the form $a\sqrt{b}$ , where a is a relative integer and b is the smallest possible integer.

$A=8\sqrt{7}-2\sqrt{28}+\sqrt{112}$

$B=2\sqrt{24}-3\sqrt{96}+9\sqrt{294}$

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