How do you simplify #(\frac { 3} { 5} ) ^ { 3} \cdot ( \frac { 3} { 4} ) ^ { - 2} \cdot ( - \frac { 5} { 4} ) ^ { 4} #?

Question

Savannah Anderson · Accepted Answer

#15/16# #=3^3*5^-3*3^2*4^-2*(-5)^4*(-4)^-4# #3*5*4^-6=15/16# Aliter : #=(3^3*3^-2*(-5)^4)/(5^3*4^-2*(-4)^4)# =#(3^(3-2)*5^(-3+4))/4^(-2+4)=(3*5)/4^2=15/16#

Avery Alexander · Answer

undefined To simplify the expression $(\frac{3}{5})^3 \cdot (\frac{3}{4})^{-2} \cdot (-\frac{5}{4})^4$, follow these steps:

1. Simplify each exponent:

$(\frac{3}{5})^3 = (\frac{3^3}{5^3}) = (\frac{27}{125})$

$(\frac{3}{4})^{-2} = \frac{1}{(\frac{3}{4})^2} = \frac{1}{(\frac{9}{16})} = \frac{16}{9}$

$(- \frac{5}{4})^4 = (\frac{-5^4}{4^4}) = (\frac{625}{256})$

2. Substitute the simplified exponents back into the expression:

$(\frac{27}{125}) \cdot (\frac{16}{9}) \cdot (\frac{625}{256})$

3. Multiply the fractions together:

$\frac{27 \cdot 16 \cdot 625}{125 \cdot 9 \cdot 256}$

4. Perform the multiplications in the numerator and denominator:

$\frac{69120}{28800}$

5. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 480:

$\frac{69120 \div 480}{28800 \div 480} = \frac{144}{60}$

6. Further simplify the fraction:

$\frac{12}{5}$

Therefore, $(\frac{3}{5})^3 \cdot (\frac{3}{4})^{-2} \cdot (-\frac{5}{4})^4$ simplifies to $\frac{12}{5}$.