Grade 8 - Square and Square Roots | Math Practice, Questions ... - Free Printable

Let's solve each problem step by step.

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Problem (1): What is the square root of 900?

The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For 900:
\[
\sqrt{900} = 30
\]
because \( 30 \times 30 = 900 \).

Answer:
\[
\boxed{30}
\]

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Problem (2): Which is the smallest number that can be used to divide 138240 to give a perfect cube?

To solve this, we need to factorize 138240 into its prime factors and then determine what additional factors are needed to make it a perfect cube.

#### Step 1: Prime factorization of 138240
\[
138240 \div 2 = 69120 \\
69120 \div 2 = 34560 \\
34560 \div 2 = 17280 \\
17280 \div 2 = 8640 \\
8640 \div 2 = 4320 \\
4320 \div 2 = 2160 \\
2160 \div 2 = 1080 \\
1080 \div 2 = 540 \\
540 \div 2 = 270 \\
270 \div 2 = 135 \\
135 \div 3 = 45 \\
45 \div 3 = 15 \\
15 \div 3 = 5 \\
5 \div 5 = 1
\]
So, the prime factorization is:
\[
138240 = 2^9 \times 3^3 \times 5
\]

#### Step 2: Make it a perfect cube
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3. Currently:
- The exponent of 2 is 9 (already a multiple of 3).
- The exponent of 3 is 3 (already a multiple of 3).
- The exponent of 5 is 1, which is not a multiple of 3. We need two more 5s to make it \( 5^3 \).

Thus, we need to divide by \( 5 \) to remove the extra factor of 5.

Answer:
\[
\boxed{5}
\]

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Problem (3): What is the square root of 0.0961?

We need to find \( \sqrt{0.0961} \). Notice that:
\[
0.0961 = \frac{961}{10000}
\]
The square root of a fraction is the square root of the numerator divided by the square root of the denominator:
\[
\sqrt{0.0961} = \sqrt{\frac{961}{10000}} = \frac{\sqrt{961}}{\sqrt{10000}}
\]
We know:
\[
\sqrt{961} = 31 \quad \text{and} \quad \sqrt{10000} = 100
\]
So:
\[
\sqrt{0.0961} = \frac{31}{100} = 0.31
\]

Answer:
\[
\boxed{d}
\]

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Problem (4): Which of the following numbers cannot be the area of a square that has an integer value side?

The area of a square with an integer side length is a perfect square. We need to check which of the given numbers is not a perfect square.

#### Check each option:
- Option (a): 3481
\[
\sqrt{3481} = 59 \quad \text{(perfect square)}
\]
- Option (b): 2116
\[
\sqrt{2116} = 46 \quad \text{(perfect square)}
\]
- Option (c): 3114
\[
\sqrt{3114} \approx 55.8 \quad \text{(not a perfect square)}
\]
- Option (d): 3136
\[
\sqrt{3136} = 56 \quad \text{(perfect square)}
\]

The number 3114 is not a perfect square.

Answer:
\[
\boxed{c}
\]

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Problem (5): Which of the following numbers is a perfect square?

We need to check which of the given numbers is a perfect square.

#### Check each option:
- Option (a): 5335
\[
\sqrt{5335} \approx 73.04 \quad \text{(not a perfect square)}
\]
- Option (b): 1846
\[
\sqrt{1846} \approx 42.96 \quad \text{(not a perfect square)}
\]
- Option (c): 1770
\[
\sqrt{1770} \approx 42.07 \quad \text{(not a perfect square)}
\]
- Option (d): 576
\[
\sqrt{576} = 24 \quad \text{(perfect square)}
\]

The number 576 is a perfect square.

Answer:
\[
\boxed{d}
\]

---

Problem (6): There are two numbers such that the sum of the numbers is 48 and their difference is 8. Find the difference of their squares.

Let the two numbers be \( x \) and \( y \) such that:
\[
x + y = 48 \quad \text{and} \quad x - y = 8
\]

#### Step 1: Solve for \( x \) and \( y \)
Add the two equations:
\[
(x + y) + (x - y) = 48 + 8 \\
2x = 56 \\
x = 28
\]
Subtract the second equation from the first:
\[
(x + y) - (x - y) = 48 - 8 \\
2y = 40 \\
y = 20
\]

So, the numbers are \( x = 28 \) and \( y = 20 \).

#### Step 2: Find the difference of their squares
The difference of squares formula is:
\[
x^2 - y^2 = (x + y)(x - y)
\]
Substitute the values:
\[
x^2 - y^2 = (28 + 20)(28 - 20) = 48 \times 8 = 384
\]

Answer:
\[
\boxed{b}
\]

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Problem (7): There are 3590 soldiers in a platoon. The commander wants to arrange them in a square formation. What is the minimum number of soldiers he needs additionally to do this kind of formation?

A square formation requires the total number of soldiers to be a perfect square. We need to find the smallest perfect square greater than or equal to 3590.

#### Step 1: Find the square root of 3590
\[
\sqrt{3590} \approx 59.91
\]
The next whole number is 60. So, the smallest perfect square greater than 3590 is:
\[
60^2 = 3600
\]

#### Step 2: Calculate the additional soldiers needed
The additional soldiers required are:
\[
3600 - 3590 = 10
\]

Answer:
\[
\boxed{b}
\]

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Problem (8): There are two squares, and the ratio of their perimeters is 8:19. If the area of the first square is 1600, then what is the area of the second one?

#### Step 1: Find the side length of the first square
The area of the first square is 1600. Let the side length be \( s_1 \):
\[
s_1^2 = 1600 \\
s_1 = \sqrt{1600} = 40
\]

#### Step 2: Use the perimeter ratio to find the side length of the second square
The perimeter of a square is \( 4 \times \text{side length} \). The ratio of the perimeters is 8:19, so the ratio of the side lengths is also 8:19. Let the side length of the second square be \( s_2 \):
\[
\frac{s_1}{s_2} = \frac{8}{19} \\
\frac{40}{s_2} = \frac{8}{19} \\
s_2 = 40 \times \frac{19}{8} = 5 \times 19 = 95
\]

#### Step 3: Find the area of the second square
The area of the second square is:
\[
s_2^2 = 95^2 = 9025
\]

Answer:
\[
\boxed{b}
\]

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Final Answers:

1. \(\boxed{30}\)
2. \(\boxed{5}\)
3. \(\boxed{d}\)
4. \(\boxed{c}\)
5. \(\boxed{d}\)
6. \(\boxed{b}\)
7. \(\boxed{b}\)
8. \(\boxed{b}\)