Question

(81÷16)^(-3÷4) × (81÷16)^(-3÷2) ÷ (5÷2)^-3
Please solve using laws if exponents

Answer 1

Laws of exponents :-

→ x^1 = x

→ x^0 = 1

→ x^(-1) = 1/x

→ x^m*x^n = x^(m+n)

→ x^m/x^n = x^(m-n)

→ (x^m)^n = x^(mn)

→ (xy)^n = x^n*y^n

→ (x/y)^n = x^n/y^n

→ x^(-n) = (1/x)^n

Solution :-

→ (81÷16)^(-3÷4) × (81÷16)^(-3÷2) ÷ (5÷2)^-3

→ [ (81/16)^(-3/4) ] × [ (81/16)^(-3/2) ] ÷ (5/2)^(-3)

using x^(-n) = 1/x^n , we get,

→ [ (16/81)^(3/4) ] × [ (16/81)^(3/2) ] ÷ (2/5)^(3)

→ [ {(2/3)⁴}^(3/4)] × [ {(4/9)²}^(3/2)] ÷ (2/5)^(3)

using (x^m)^n = x^(mn) now, we get,

→ [ (2/3)³ ] × [ (4/9)³ ] ÷ (2/5)³

→ (8/27) × (64/729) ÷ (8/125)

→ (8/27) × (64/729) × (125/8)

→ ( 8 * 64 * 125) / ( 27 * 729 * 8)

→ ( 64 * 125) / (27 * 729)

→ (4³ * 5³) / (3³ * 9³)

using , (x/y)^n = x^n/y^n now,

→ [ (4 * 5) / (3 * 9) ] ³

→ (20/27)³ Or, (8000/19683) (Ans.)

Answer 2

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$\huge\tt{SOLUTION:}$

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See the attachments...

As we are known to the laws of exponents,we can solve the following as..

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$\huge\tt{SOLVING:}$

➠(81÷16)^(3÷4)×(81÷16)^(-3÷2)÷(5÷2)^-3

➠[(81/16)^(-3/4)]×[(81/16)^(-3/2)]÷{5/2}^(-3)

➠[(16/81)^(3/4)]×[(16/81)^(3/2)]÷{2/5)^(-3)

➠[{(2/3)⁴}^(3/4)}×{4/9²}^(3/2)]÷(2/5)^(3)

➠[(2/3)³]×[4/9)³÷{2/5}³

➠(8/27)×(64/729)×(125/8)

➠(8×64×125)/(27×729×8)

➠(64×125)/(27×729)

➠[(64×125)/(27×729)]

➠(8000/19683)

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