Chapter 3: Squares and Square Roots & Cubes and Cube Roots

Solution:-

First we have to find out the LCM of 8, 9 and 10

So, LCM of 8, 9, and 10 = 2 × 2 × 2 × 3 × 3 × 5

= 360

Now, grouping the factors, (2× 2) × 2 × (3 × 3) × 5

i.e. 2 and 5 is not able to make their pair.

So, to make it perfect square, 360 must be multiplied with 2 × 5 = 10

Then,

= 360 × 10 = 3600

∴ 3600 is least square number which is exactly divisible by 8, 9, and 10.

Solution:-

From the question it is given that,

Area of a square plot =
m² = 40401/400

We know that, area of square plot = side²

(40401/400) = side²

By taking square root on both side,

Side = √(40401/400)

= √((201 × 201)/(20 × 20))

= √(201²/20²)

= 201/20

=
m

∴the length of one side of the plot is
m

Solution:-

From the question it is given that 324

Now, we subtract successive odd numbers starting from 1 as:

324 – 1 = 323

323 – 3 = 320

320 – 5 = 315

315 – 7 = 308

308 – 9 = 299

299 – 11 = 288

288 – 13 = 275

275 – 15 = 260

260 – 17 = 243

243 – 19 = 224

224 – 21 = 203

203 – 23 = 180

180 – 25 = 155

155 – 27 = 128

128 – 29 = 99

99 – 31 = 68

68 – 33 = 35

35 – 35 = 0

Here we see 324 reduces to 0 after subtracting 18 odd numbers.

Therefore the square root of 324 is 18.

Solution:-

Let us assume the three number be 2a, 3a, 4a

Then,

Given, sum of cube of three numbers is 0.334125

i.e. (2a)³ + (3a)³ + (4a)³ = 0.334125

8a³ + 27a³ + 64a³ = 0.334125

99a³ = 0.334125

a³ = 0.334125/99

a³ = 0.003375

a³ = 3375/1000000

a = ³√(3375/1000000)

a = ³√((15 × 15 × 15)/(10 × 10 × 10 × 10 × 10 × 10))

a = ³√(15³/10⁶)

a = 15/1000

a = 0.015

∴The numbers are, 2a = 2 × 0.015 = 0.03

3a = 3 × 0.015 = 0.045

4a = 4 × 0.015 = 0.06

Solution:-

= ³√27 + ³√0.008 + ³√0.064

= ³√(3 × 3 × 3) + (³√(8/1000)) + ³√(64/1000)

= ³√(3³) + (³√(2 × 2 × 2)/(10 × 10 × 10)) + ³√((4 × 4 × 4)/(10 × 10 × 10))

= 3 + (³√(2³/10³) + ³√(4³/10³)

= 3 + (2/10) + (4/10)

= 3 + 0.2 + 0.4

= 3.6

Solution:-

Given, {(5² + (12²)^1/2)}³

= {(25 + (144)^1/2)}³

= {(25 + (12)}³

= {37}³

= 37 × 37 × 37

= 50,653

Solution:-

Given, {(6² + (8²)^1/2)}³

= {(36 + (64)^1/2)}³

= {(36 + (8)}³

= {44}³

= 44 × 44 × 44

= 85,184

Solution:-

Let us assume PQRS is perfect square,

Where, P = even, Q = even, R = odd, S = even

So, 8836 is the perfect square.

Solution:-

6, 19 and 30 are the three numbers in which, when we add the end of each line we always get a perfect square.

6 + 19 = 25

6 + 30 = 36

19 + 30 = 49

Solution:-

From the question it is given that,

Perimeters of first squares = 40m

Perimeter of second square = 96m

Let us assume side of first square = x₁

and assume side of second square = x₂

WKT, perimeter of first square = 4 × side

40 = 4x₁

X₁ = 40/4

X₁ = 10 m

Then,

Perimeter of second square = 4 × side

96 = 4x₂

X₂ = 96/4

X₂ = 24 m

As per the condition given in the question area of another square is equal to the first two squares.

Let us assume y² be the area of another square.

So,

y² = x₁² + x₂²

y² = 10² + 24²

y² = 100 + 576

y = √(676)

y = 26 m

Then,

Perimeter of another square = 4 × side

= 4 × y

= 4 × 26

= 104 m