b) $$ \frac{\frac{2}{9}-\left(\frac{3}{2}\right)^{-1}}{5^{0}-5: 4^{-2}} $$

Question

UpStudy AI · Accepted Answer

Let's evaluate the expression step by step:

$$
b) \quad \frac{\frac{2}{9} - \left(\frac{3}{2}\right)^{-1}}{5^{0} - 5 \div 4^{-2}}
$$

**Step 1: Simplify the Numerator**

$$
\frac{2}{9} - \left(\frac{3}{2}\right)^{-1}
$$

Recall that $$ \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} $$. Therefore:

$$
\left(\frac{3}{2}\right)^{-1} = \frac{2}{3}
$$

Now, substitute back into the numerator:

$$
\frac{2}{9} - \frac{2}{3} = \frac{2}{9} - \frac{6}{9} = -\frac{4}{9}
$$

**Step 2: Simplify the Denominator**

$$
5^{0} - 5 \div 4^{-2}
$$

First, evaluate each term:

- $$ 5^{0} = 1 $$
- $$ 4^{-2} = \frac{1}{4^{2}} = \frac{1}{16} $$

Now, perform the division:

$$
5 \div 4^{-2} = 5 \div \left(\frac{1}{16}\right) = 5 \times 16 = 80
$$

Substitute back into the denominator:

$$
1 - 80 = -79
$$

**Step 3: Combine Numerator and Denominator**

$$
\frac{-\frac{4}{9}}{-79} = \frac{4}{9} \times \frac{1}{79} = \frac{4}{711}
$$

**Final Answer:**

$$
\boxed{\dfrac{4}{711}}
$$
The final answer is $$ \frac{4}{711} $$.

b)