SQUARE ROOTS & CUBE ROOTS

KEY STAGE 3 BINARY NUMBERSBinary is the language used by computers. It uses 0 and 1 to represent different numbers.In the everyday number system, we use 0-9 to show numbers.

If a number has two whole numbers to its right, we know that it has a ‘hundreds’ value (H). For example, the 5 above represents 500. This number system is known as ‘Base 10’. Each position to the left is worth 10× more than the place to the right:

Binary uses ‘Base 2’. Each position to the left is worth 2× more than the place to the right:

Of course you don’t have to write in 1, 2, 4, etc. You just remember them like you remember HTU (Hundreds, Tens, Units).

H T U5 6 9

100

16 8 4 2 1

10

x10

x2 x2 x2 x2

x10

1

√ is the square root sign. Taking the square root is the opposite of squaring. When a number is square rooted it has two square roots, one positive and one negative.For example:√25 = 5 or –5 since (5)² = 25 and (–5)² = 25√196 = √(4 × 49) = 2 × 7 = 14

It is important to note that √a + √b is not equal to √a+bFor example, √9 + √4 is not equal to √13

A surd is the square root of any number that is not a square number. It cannot be written exactly as a decimal.For example, √2, √3, √5, √6, √7, … are allsurds.

Example:Write √18 in terms of the simplest possible surd:√18 = √9 x √2= 3 x √2= 3 √2

1. Find the value of:

a √25 b ³√64 c √144

d ³√–64 e 3² f 4³ g 104

2. Write the following numbers in binary.

a 12 b 23 c 16

³√ is the cube root sign. Taking thecube root is the opposite of cubing.Examples:³√27 = 3 since 3 × 3 × 3 = 27³√–125 = –5 since –5 × –5 × –5 = –125

√ is known as the fourth root.Examples:√16 = 2since 2 × 2 × 2 × 2 = 16

√ is known as the fifth root.Examples:√243 = 3since 3 × 3 × 3 × 3 × 3 = 243