**Simplify the following using laws of indices:**

**(i) (16)**^{-0.75 }**x ****(64)**^{4/3 }

(16)^{-0.75 }x (64)^{4/3 }[16 = 2^{4} ]

= (24)^{-0.75 }x (24)^{4/3 }[64 = 2^{6} ]

= 2^{4} x ^{-3/4 }x 2^{6} x ^{4/3 }

= 2^{-3 }x 2^{8}

= 2^{5}

**= 32**

**(ii) (0.25)**^{0.5}**x ****(100)**^{-1/2}

(0.25)^{0.5} x (100)^{-1/2}

= (0.25)^{1/2} x (^{1}/_{100} )^{1/2}

= (0.25)^{1/2} x (^{1}/_{10}^{2} )^{1/2}

= (0.5) x ( ^{1}/_{10} )

= ( ^{5}/_{10} ) x ( ^{1}/_{10})

= ^{5}/_{100}

**= **^{𝟏}/_{𝟐𝟎}

**(iii) (6.25)**^{0.5}**x ****10**^{2}**x ****(100)**^{-1/2}**x ****(0.01)**^{-1}

= ( ^{625}/_{100} )^{1/2} x 10^{2} x (^{1}/_{10})^{1/2} x ( ^{1}/_{100} )^{-1}

= ( ^{25}/_{10} )^{1/2} x 10^{2} x (^{1}/_{10} )^{1/2} x (^{1}/_{100})^{-1}

= ( 25^{2}/_{10} )^{1/2} x 10^{2} x (^{1}/_{10}^{2} )^{1/2} x (100)^{1}

= ( ^{25}/_{10} ) x 100 x 1 10 x 100

**= 2500 **

**(iv) (3**^{-1/2}**x ****2**^{-1/3}**) ÷ (3**^{-3/4}**x ****2**^{-5/6}**) **

= 3^{−1/2}× 2^{−1/3} 3^{−3/4}× 2^{−5/6}

= 3^{-1/2+3/4} x 2^{-1/3+5/6 }

= 3^{1/4 }x 2^{3/6 }

= 3^{1/4 }x 2^{1/2 }

= 3^{1/4 }x 2^{1/4 }

=(3 × 2^{2})^{1/4}

= (3 × 4)^{1/4}

**= **𝟏𝟐^{1/4}

**Find the value of the expression.**

**[ 3 ^{1/3 }{5^{-1/2 }x 3^{-1/3 }x (225^{2})^{1/3}}^{1/2}]^{6}**

Solution:

= [ 3^{1/3 }{5^{-1/2}^{x}^{-1/2 }x 3^{-1/3}^{x}^{1/2 }x (225^{2})^{2/3}^{x}^{1/2}]^{6}

= [ 3^{1/3 }{5^{-1/4 }x 3^{1/6 }x (225^{2})^{-2/6}]^{6}

= [ 3^{1/3 }^{x }^{6 }{5^{1/4 }^{x }^{6 }x 3^{1/6 }^{x }^{6 }x (225^{2})^{-2/6 }^{x }^{6}]

= [ 32 x 5^{1/4 }^{x }^{6 }x 3^{1/6 }^{x }^{6 }x 225^{2-2/6 }^{x }^{6}]

= [32 x 5^{3/2} x 3^{1} x 225^{-2}]

= [32 x 5^{2/3} x 3^{1 }x 15^{-4}]

= [32 x 5^{-2/3 }x 31 x 3^{-4} x 5^{-4}]

= [3^{2 + 1 – 4 }x 5^{3/2} – 4]

= ^{1}/_{3}^{1} x ^{1}/_{5}^{5/2}

= ^{1}/_{3}^{1} x ^{1}/_{5}^{5/2}

= ^{1}/_{3}^{1} x ^{1}/_{√5}^{5}

= ^{1}/_{3} x ^{1}/_{√3125}

= ^{1}/_{3√3125}

**Simplify:**

**[{(3**^{5/2 }**x ****5**^{3/4}**) ÷ 2**^{-5/4}**} ÷ {16 / (5**^{2}**x ****2**^{1/4}**x ****3**^{1/2}**)}]**^{1/5}

Solution:

= [{(3^{5/2} × 5^{3/4}) ÷ 2^{−5/4}} ÷ {16÷ (5^{2}× 2^{1/4}× 3^{1/2})}] ^{1/5 }

= [{(3^{5/2} × 5^{3/4})/(_{2}^{−54}) ÷ ^{16}/(_{5}^{2}× _{2}^{1/4}× _{3}^{1/2})}] ^{1/5}

= [(3^{5/2} × 5^{3/4})/(_{2}^{−54}) ÷ (5^{2}× 2^{1/4}× 3^{1/2})/_{2}^{4}]^{1/5}

= [(3^{5/2} × 5^{3/4} × 5^{2/1} × 2^{1/4} × 3^{1/2})/(_{2}^{−5/4} × _{2}^{4 })]^{1/5}

= [2^{1/4} × 3^{5/2 + 1/2} × 5^{3/4 + 2/1}/_{2}^{−5/4 + 4/1}]^{1/5}

= 2^{1/4} × 3^{5/2 + 1/2} × 5^{11/4}/_{2}^{11/4}]^{1/5}

= [2^{1/4 }^{– }^{4/11} × 3^{3} × 5^{11/4 }] ^{1/5}

= [2^{10/4 × 1/5} × 3^{3 }^{× 1/5} × 5^{11/4 }^{+ }^{1/5}]

= 2^{−}^{1/2} × 3^{1/5} × 5^{11/20}