Find each of the following in its simplest form: 1) (3/5)^3, 2) (-7)^4, 3) (9/4) x (2/3)^2, 4) 2 * (7/9) ÷ (-1 2/3)^2, 5) (-2/5)^2 x (-5/2)^3, 6) (-2)^4 + (-3)^3.

Steps to Solve

1. Simplify ((\frac{3}{5})^3)

To simplify ((\frac{3}{5})^3), we need to cube both the numerator and the denominator:

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

2. Calculate ((-7)^4)

Calculating the fourth power of (-7):

$$ (-7)^4 = 7^4 = 2401 $$

3. Simplify ((\frac{9}{4}) \times (\frac{2}{3})^2)

First, calculate ((\frac{2}{3})^2):

$$ \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} $$

Now, multiply by (\frac{9}{4}):

$$ \frac{9}{4} \times \frac{4}{9} = 1 $$

4. Calculate (2 \times (\frac{7}{9}) \div (-\frac{5}{3})^2)

First, calculate ((-1 \frac{2}{3})^2):

Convert mixed number (-1 \frac{2}{3}) to an improper fraction:

$$ -\frac{5}{3} = -\frac{5}{3} $$

Then square it:

$$ \left(-\frac{5}{3}\right)^2 = \frac{25}{9} $$

Now the expression becomes:

$$ 2 \times \frac{7}{9} \div \frac{25}{9} = 2 \times \frac{7}{9} \times \frac{9}{25} = \frac{14}{25} $$

5. Simplify ((-2/5)^2 \times (-5/2)^3)

First, calculate ((-2/5)^2):

$$ \left(-\frac{2}{5}\right)^2 = \frac{4}{25} $$

Now calculate ((-5/2)^3):

$$ \left(-\frac{5}{2}\right)^3 = -\frac{125}{8} $$

Now multiply both results together:

$$ \frac{4}{25} \times -\frac{125}{8} = -\frac{500}{200} = -\frac{5}{2} $$

6. Calculate ((-2)^4 + (-3)^3)

Calculate ((-2)^4):

$$ (-2)^4 = 16 $$

Now calculate ((-3)^3):

$$ (-3)^3 = -27 $$

Now add both results together:

$$ 16 + (-27) = -11 $$