krit.club logo

Number System - Squares, cubes, and roots

Grade 6IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Square Numbers: The result of multiplying an integer by itself (e.g., 3imes3=93 imes 3 = 9).

•

Square Roots: The inverse operation of squaring; finding the number that was multiplied by itself to get the square.

•

Perfect Squares: Numbers whose square roots are whole numbers (e.g., 1, 4, 9, 16, 25, 36, 49, 64, 81, 100).

•

Cube Numbers: The result of multiplying a number by itself three times (e.g., 2imes2imes2=82 imes 2 imes 2 = 8).

•

Cube Roots: The inverse operation of cubing; finding the number that was multiplied by itself twice to get the cube.

•

Inverse Operations: Squaring and square rooting are opposites, as are cubing and cube rooting.

•

Order of Operations: Squares and roots (Indices/Orders) are calculated after Brackets but before Division/Multiplication (BIDMAS/BODMAS).

📐Formulae

n2=n×nn^2 = n \times n

x=n\sqrt{x} = n if n2=xn^2 = x

n3=n×n×nn^3 = n \times n \times n

x3=n\sqrt[3]{x} = n if n3=xn^3 = x

(ab)2=a2×b2(ab)^2 = a^2 \times b^2

💡Examples

Problem 1:

Evaluate 72+647^2 + \sqrt{64}

Solution:

49+8=5749 + 8 = 57

Explanation:

First, calculate the square of 7 (7×7=497 \times 7 = 49). Then, find the square root of 64, which is 8 because 8×8=648 \times 8 = 64. Finally, add the results together.

Problem 2:

Find the value of 43−2734^3 - \sqrt[3]{27}

Solution:

64−3=6164 - 3 = 61

Explanation:

First, calculate the cube of 4 (4×4×4=644 \times 4 \times 4 = 64). Then, find the cube root of 27, which is 3 because 3×3×3=273 \times 3 \times 3 = 27. Subtract 3 from 64.

Problem 3:

A square garden has an area of 121 m2121 \text{ m}^2. What is the length of one side?

Solution:

121=11 m\sqrt{121} = 11 \text{ m}

Explanation:

The area of a square is calculated by side2\text{side}^2. To find the length of one side, find the square root of the area (11×11=12111 \times 11 = 121).

Problem 4:

Estimate the value of 40\sqrt{40} to the nearest whole number.

Solution:

66

Explanation:

Identify the perfect squares surrounding 40. These are 3636 (626^2) and 4949 (727^2). Since 40 is much closer to 36 than to 49, the square root is approximately 6.