Finding Square and Nth Roots Worksheet | Printable Maths Worksheets - Free Printable

Problem Overview:

The task involves calculating squares, cubes, higher powers, square roots, cube roots, and nth roots both without and with a calculator. Let's solve each section step by step.

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Section 1: Without a Calculator

#### Part 1: Squares
We need to calculate the squares of numbers and find the square roots of given numbers.

| \( 1^2 \) | \( 2^2 \) | \( 3^2 \) | \( 4^2 \) | \( ?^2 = 81 \) | \( ?^2 = 36 \) | \( ?^2 = 144 \) | \( ?^2 = 25 \) |
|-----------|-----------|-----------|-----------|----------------|----------------|-----------------|----------------|
| 1 | 4 | 9 | 16 | 9 | 6 | 12 | 5 |

- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( ?^2 = 81 \): The square root of 81 is 9.
- \( ?^2 = 36 \): The square root of 36 is 6.
- \( ?^2 = 144 \): The square root of 144 is 12.
- \( ?^2 = 25 \): The square root of 25 is 5.

#### Part 2: Cubes
We need to calculate the cubes of numbers and find the cube roots of given numbers.

| \( 1^3 \) | \( 2^3 \) | \( 3^3 \) | \( 4^3 \) | \( ?^3 = 1000 \) | \( ?^3 = -1 \) | \( ?^3 = 125 \) | \( ?^3 = 0.008 \) |
|-----------|-----------|-----------|-----------|------------------|----------------|-----------------|-------------------|
| 1 | 8 | 27 | 64 | 10 | -1 | 5 | 0.2 |

- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( ?^3 = 1000 \): The cube root of 1000 is 10.
- \( ?^3 = -1 \): The cube root of -1 is -1.
- \( ?^3 = 125 \): The cube root of 125 is 5.
- \( ?^3 = 0.008 \): The cube root of 0.008 is 0.2 (since \( 0.2 \times 0.2 \times 0.2 = 0.008 \)).

#### Part 3: Higher Powers
We need to calculate various powers of numbers.

| \( 5^4 \) | \( 3^4 \) | \( 2^5 \) | \( 6^3 \) | \( 11^2 \) | \( (0.1)^3 \) | \( (-3)^2 \) | \( \left( \frac{1}{2} \right)^2 \) |
|-----------|-----------|-----------|-----------|------------|---------------|---------------|-----------------------------|
| 625 | 81 | 32 | 216 | 121 | 0.001 | 9 | 0.25 |

- \( 5^4 = 5 \times 5 \times 5 \times 5 = 625 \)
- \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \)
- \( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \)
- \( 6^3 = 6 \times 6 \times 6 = 216 \)
- \( 11^2 = 11 \times 11 = 121 \)
- \( (0.1)^3 = 0.1 \times 0.1 \times 0.1 = 0.001 \)
- \( (-3)^2 = (-3) \times (-3) = 9 \)
- \( \left( \frac{1}{2} \right)^2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 0.25 \)

#### Part 4: Roots
We need to calculate square roots, cube roots, and higher roots.

| \( \sqrt{16} \) | \( \sqrt{49} \) | \( \sqrt{121} \) | \( \sqrt{64} \) | \( \sqrt[3]{8} \) | \( \sqrt[3]{216} \) | \( \sqrt[4]{81} \) | \( \sqrt[5]{-1} \) |
|-----------------|-----------------|------------------|-----------------|-------------------|---------------------|--------------------|---------------------|
| 4 | 7 | 11 | 8 | 2 | 6 | 3 | -1 |

- \( \sqrt{16} = 4 \)
- \( \sqrt{49} = 7 \)
- \( \sqrt{121} = 11 \)
- \( \sqrt{64} = 8 \)
- \( \sqrt[3]{8} = 2 \) (since \( 2 \times 2 \times 2 = 8 \))
- \( \sqrt[3]{216} = 6 \) (since \( 6 \times 6 \times 6 = 216 \))
- \( \sqrt[4]{81} = 3 \) (since \( 3 \times 3 \times 3 \times 3 = 81 \))
- \( \sqrt[5]{-1} = -1 \) (since \( -1 \times -1 \times -1 \times -1 \times -1 = -1 \))

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Section 2: With a Calculator

We need to use a calculator to compute the following expressions:

| \( 6.3^4 \) | \( \left( \frac{4}{9} \right)^5 \) | \( \left( -2 \frac{3}{7} \right)^6 \) | \( \left( \frac{3 \times 46}{9.5 \times 4} \right)^4 \) | \( \sqrt[3]{500} \) | \( \sqrt[5]{-64.09} \) | \( \sqrt[4]{\frac{13}{200}} \) | \( \sqrt[3]{\left( \frac{7.5^2}{\frac{4}{11}} \right)} \) |
|-------------|------------------------------------|---------------------------------------|------------------------------------------------------|---------------------|-------------------------|--------------------------------|----------------------------------------------------------|
| ≈ 1575.29 | ≈ 0.0132 | ≈ 1176.49 | ≈ 1.0000 | ≈ 7.937 | ≈ -2.299 | ≈ 0.397 | ≈ 10.392 |

- \( 6.3^4 \approx 1575.29 \)
- \( \left( \frac{4}{9} \right)^5 \approx 0.0132 \)
- \( \left( -2 \frac{3}{7} \right)^6 = \left( -\frac{17}{7} \right)^6 \approx 1176.49 \)
- \( \left( \frac{3 \times 46}{9.5 \times 4} \right)^4 = \left( \frac{138}{38} \right)^4 = 3.6316^4 \approx 1.0000 \)
- \( \sqrt[3]{500} \approx 7.937 \)
- \( \sqrt[5]{-64.09} \approx -2.299 \)
- \( \sqrt[4]{\frac{13}{200}} \approx 0.397 \)
- \( \sqrt[3]{\left( \frac{7.5^2}{\frac{4}{11}} \right)} = \sqrt[3]{\left( \frac{56.25 \times 11}{4} \right)} = \sqrt[3]{151.875} \approx 10.392 \)

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Final Answer

Combining all the results:

Without a Calculator:
- Squares: \( 1, 4, 9, 16, 9, 6, 12, 5 \)
- Cubes: \( 1, 8, 27, 64, 10, -1, 5, 0.2 \)
- Higher Powers: \( 625, 81, 32, 216, 121, 0.001, 9, 0.25 \)
- Roots: \( 4, 7, 11, 8, 2, 6, 3, -1 \)

With a Calculator:
- \( 6.3^4 \approx 1575.29 \)
- \( \left( \frac{4}{9} \right)^5 \approx 0.0132 \)
- \( \left( -2 \frac{3}{7} \right)^6 \approx 1176.49 \)
- \( \left( \frac{3 \times 46}{9.5 \times 4} \right)^4 \approx 1.0000 \)
- \( \sqrt[3]{500} \approx 7.937 \)
- \( \sqrt[5]{-64.09} \approx -2.299 \)
- \( \sqrt[4]{\frac{13}{200}} \approx 0.397 \)
- \( \sqrt[3]{\left( \frac{7.5^2}{\frac{4}{11}} \right)} \approx 10.392 \)

\boxed{
\begin{array}{c}
\text{Without a Calculator:} \\
\text{Squares: } 1, 4, 9, 16, 9, 6, 12, 5 \\
\text{Cubes: } 1, 8, 27, 64, 10, -1, 5, 0.2 \\
\text{Higher Powers: } 625, 81, 32, 216, 121, 0.001, 9, 0.25 \\
\text{Roots: } 4, 7, 11, 8, 2, 6, 3, -1 \\
\\
\text{With a Calculator:} \\
6.3^4 \approx 1575.29, \left( \frac{4}{9} \right)^5 \approx 0.0132, \left( -2 \frac{3}{7} \right)^6 \approx 1176.49, \left( \frac{3 \times 46}{9.5 \times 4} \right)^4 \approx 1.0000, \\
\sqrt[3]{500} \approx 7.937, \sqrt[5]{-64.09} \approx -2.299, \sqrt[4]{\frac{13}{200}} \approx 0.397, \sqrt[3]{\left( \frac{7.5^2}{\frac{4}{11}} \right)} \approx 10.392
\end{array}
}