Question

Without using a calculator show that (49/16) raised to the power of negative 3/2 equals 64/343

show all steps ​

Answer 1

Explanation:

Apply exponent rule
`a^(-b)=(1)/(a^b)`:

`\implies \left( (49)/(16) \right) ^{-{\frac32}}=(1)/(\left( (49)/(16) \right) ^(\frac32))`

Apply exponent rule
`\left( (a)/(b) \right) ^c=(a^c)/(b^c)`:

`\implies (1)/(\left( (49)/(16) \right) ^(\frac32))=(1)/(\left( (49^(\frac32))/(16^(\frac32)) \right) )`

Factor the 49 and 16:

`\implies (1)/(\left( (49^(\frac32))/(16^(\frac32)) \right) )=(1)/(\left( ((7^2)^(\frac32))/((4^2)^(\frac32)) \right) )`

Apply exponent rule
`(a^b)^c=a^(bc)`:

`\implies (1)/(\left( ((7^2)^(\frac32))/((4^2)^(\frac32)) \right) )= (1)/(\left( (7^3)/(4^3) \right) )=(1)/(\left( (343)/(64) \right) )`

Apply fraction rule
`(1)/((a)/(b))=(b)/(a)`

`\implies (1)/(\left( (343)/(64) \right) )=(64)/(343)`