**Write the following surds in their simplest form:**

**(i) √𝟕𝟔**

= √(19 × 4)

= √(19 ×2^{4})

**= **𝟐√𝟏𝟗

**(ii) **∛** (**𝟏𝟎𝟖)

= ∛(3^{3} × 2^{2})

= ∛2^{2}

**= **∛𝟒

**(iii) ****∜**𝟓𝟎𝟎𝟎

= **∜**5000

= **∜(**5^{11} × 8^{4})

**= **𝟓**∜**𝟖

**(iv) **∛^{𝟏𝟖𝟗}/_{𝟐𝟓}

= ∛[(3^{3} × 7)/_{5}^{2}]

**= 3**∛^{𝟕}/_{𝟐𝟓}

**(v) ****∜**^{𝟒𝟎𝟎}/_{𝟒𝟗}

= **∜(**2^{4} × 5^{2})/_{7}^{24}

= 2**∜**5^{2}/7^{2}

**= 2****∜**^{𝟐𝟓}/_{𝟒𝟗 }

**Classify the following in to like surds**

**(i) √**𝟐𝟒 **, √**𝟏𝟐𝟖 **, √**𝟕𝟓 **, √**𝟕𝟐 **, √**𝟓𝟒 **, √**𝟐𝟒

= **√**3^{5} , **√**2^{7} , **√(**5^{2}×3) , **√(**3^{2}×2^{3}) , **√(**2 × 3^{3}) , **√(**2^{3}×3)

= 3^{2}**√**3 , 2^{3}**√**2 , 5**√**3 , 3×2**√**2 , 3**√**(2×3), 2**√**(2×3)

= 9**√**3, 8**√**2, 5**√**3, 6**√**2, 3**√**6, 2**√**6

= {8**√**2, 6**√**2}, {9**√**3, 5**√**3}, {2**√**6, 3**√**6,}

**={ √**𝟏𝟐𝟖**, √**𝟕𝟐**}, { √**𝟐𝟒𝟑**, √**𝟕𝟓**}, { √**𝟓𝟒**, √**𝟐𝟒**,}**

**(ii) **∛𝟐𝟎𝟎𝟎**, **∛𝟔𝟖𝟔**, **∛𝟔𝟒𝟖**, **∛𝟑𝟕𝟓**, **∛𝟏𝟐𝟖**, **∛𝟐𝟒

= ∛(2×52), ∛(73×2), ∛(63× 3), ∛(5^{3}×3), ∛(4^{3}×2), ∛(2^{3}× 3)

= 10∛2, 7∛2, 6∛3, 5∛3, 4∛2, 2∛3

**= {4**∛𝟐**, 7**∛𝟐**, 10**∛𝟐**} & {2**∛𝟑**, 5**∛𝟑**, 6**∛𝟑**} **

**= { **∛𝟐𝟎𝟎𝟎**, **∛𝟔𝟖𝟔 **, **∛𝟏𝟐𝟖**} & { **∛𝟔𝟒𝟖**, **∛𝟑𝟕𝟓**, **∛𝟐𝟒**} **

**Which of the following are pure surds?**

(i) **√**296 = **√(**3^{3}×37) = 2**√(**37×2) = **2√**𝟕𝟒 – not pure surds

(ii) **√**729 = **√**3^{6} = 3^{3} = **27 – **not pure surds

(iii) ∛ 211 Cannot be reduced further hence it is a pure surd. **– Yes, a pure surd**

(iv) **∜** 75 is also a pure surd. **– Yes, **a pure surds

(v) ∛ 296 = ∛(2^{3}×37) = **2** ∛𝟑𝟕 – no, not a pure surd

(vi) **∜**296 = cannot be reduced further, hence it is a pure surd **– Yes **

**Write the following irrational numbers sure from**

**(i) √(**𝟏𝟓√(𝟐𝟕))

= **√**27 = **√**3^{3} = 3**√**3

= [15(27)^{1/2}]^{1/2}

= 15^{2/4} x 27^{1/4} [^{1}/_{2} = ^{4}/_{2}]

= ∜(15^{3} × 27)

= ∜(225 × 27)

**= **∜𝟔𝟎𝟕𝟓

** **

**(ii) √(**𝟒𝟎(**√**𝟏𝟐))

= [40 × (12)^{1/2}] ^{½}

= (40)^{1/2} × (12)^{1/4}

= ∜(40^{2} × (12)^{1})

= ∜(40^{2}× 121)

= ∜(1600×12)

= ∜19200

**(iii) √(**𝟓(**√**𝟒𝟖))

= [5^{1/2} ×(48)^{1/2}]^{½}

= [5^{1/2} × 48^{1/4}]

= ∜ (5^{2}× 48)

= ∜(25×48)

= ∜1200

**5. Reduce the following to surds of the same order. **

**(i) ****∛** **, √**𝟐**and **𝟓^{1/5}

The orders are 3, 2, and 5

LCM of 2, 3 and 5 is 30

∛2 = 2^{1/3 }= 2^{10/30} = (𝟐^{𝟏𝟎})^{ 1/3𝟎}

**√**2 = 2^{1/2 }= 2^{15/30} = (𝟐^{𝟏𝟓 })^{ 1/3𝟎}

5^{1/5} = 5^{1/5} = 5^{6/30} = (𝟓^{𝟔}) ^{1/3𝟎}

(1024)^{1/3𝟎} , (32768)^{ 1/3𝟎} , (15625)^{ 1/3𝟎}

They have the same order 30

**(ii) √**𝟓 **, (√**𝟏𝟓)^{1/4} **, and (√**𝟓𝟎)^{1/8}

Order is 2, 4, 8

Their LCM is 8

**√**5 = 5^{1/2} = 5^{4/8} = (5^{4 })^{1/8} = (125)^{1/8}

**(√**15)^{1/4} = (**√**15)^{1/4} = (15)^{2/8} = ((15)^{2})^{ 1/8} = (225)^{1/8}

(50)^{1/8} is in its simplest form

(50)^{1/8} **, **𝟐𝟐𝟓𝟖 **, **𝟏𝟐𝟓𝟖

∴ Thus they all are of the same order 8.

**(iii) √**𝟐 **, √**𝟕^{1/3} **, (√**𝟏𝟏)^{1/4} **and (√**𝟏𝟔𝟕𝟏)^{𝟏/1𝟐}

Order is 2, 3, 4, and 12

Their LCM is 12

**√**2 = 2^{1/2} = 2^{6/12} = (**√**2^{6})^{1/12} = (64)^{1/12}

**√**7^{1/3} = 7^{4/12} = (7^{4} )^{1/12} = (2401)^{1/12}

**(√**11)^{1/4} = (11)^{3/12} = ((11)^{3})^{1/12} = (1331)^{1/12}

167112 is in its simplest form

𝟔𝟒^{𝟏/1𝟐} **, **𝟐𝟒𝟎𝟏^{𝟏/1𝟐} **, **𝟏𝟑𝟑𝟏^{𝟏/1𝟐} **, **𝟏𝟔𝟕𝟏^{𝟏/1𝟐}

∴ Thus they all are of the same order 12

Pingback: IX – Table of Contents – Breath Math