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

Answer:

Step-by-step explanation:

Apply exponent rule  $a^{-b}=\dfrac{1}{a^b}$:

$\implies \left( \dfrac{49}{16} \right) ^{-{\frac32}}=\dfrac{1}{\left( \dfrac{49}{16} \right) ^{\frac32}}$

Apply exponent rule $\left( \dfrac{a}{b} \right) ^c=\dfrac{a^c}{b^c}$:

$\implies \dfrac{1}{\left( \dfrac{49}{16} \right) ^{\frac32}}=\dfrac{1}{\left( \dfrac{49^{\frac32}}{16^{\frac32}} \right) }$

Factor the 49 and 16:

$\implies \dfrac{1}{\left( \dfrac{49^{\frac32}}{16^{\frac32}} \right) }=\dfrac{1}{\left( \dfrac{(7^2)^{\frac32}}{(4^2)^{\frac32}} \right) }$

Apply exponent rule $(a^b)^c=a^{bc}$:

$\implies \dfrac{1}{\left( \dfrac{(7^2)^{\frac32}}{(4^2)^{\frac32}} \right) }= \dfrac{1}{\left( \dfrac{7^3}{4^3} \right) }=\dfrac{1}{\left( \dfrac{343}{64} \right) }$

Apply fraction rule $\dfrac{1}{\frac{a}{b}}=\dfrac{b}{a}$

$\implies \dfrac{1}{\left( \dfrac{343}{64} \right) }=\dfrac{64}{343}$