The expression you provided is:

$-\frac{2}{5} + \left( \frac{4}{9} + \left( -\frac{3}{4} \right) \right)$

Let's solve it step by step.

Step 1: Simplify the Inner Parentheses

Start by simplifying $\frac{4}{9} + \left(-\frac{3}{4}\right)$.

To add these two fractions, we need a common denominator:

• The least common multiple of 9 and 4 is 36.

Rewrite each fraction with a denominator of 36: $\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}$ $-\frac{3}{4} = -\frac{3 \times 9}{4 \times 9} = -\frac{27}{36}$

Now we can add these two fractions: $\frac{16}{36} + \left(-\frac{27}{36}\right) = \frac{16 - 27}{36} = \frac{-11}{36}$

Step 2: Substitute Back and Continue

Now, substitute $\frac{-11}{36}$ back into the original expression: $-\frac{2}{5} + \frac{-11}{36}$

Step 3: Add $-\frac{2}{5}$ and $-\frac{11}{36}$

To add these, find a common denominator:

• The least common multiple of 5 and 36 is 180.

Rewrite each fraction with a denominator of 180: $-\frac{2}{5} = -\frac{2 \times 36}{5 \times 36} = -\frac{72}{180}$ $-\frac{11}{36} = -\frac{11 \times 5}{36 \times 5} = -\frac{55}{180}$

Now add these two fractions: $-\frac{72}{180} + \left(-\frac{55}{180}\right) = -\frac{72 + 55}{180} = -\frac{127}{180}$

Final Answer

$-\frac{2}{5} + \left( \frac{4}{9} + \left( -\frac{3}{4} \right) \right) = -\frac{127}{180}$

Would you like further clarification or have any questions?

Related Questions

1. How do you find a common denominator for adding fractions with different denominators?
2. What steps are involved in simplifying nested parentheses in an expression?
3. How can fractions with negative values be combined effectively?
4. What techniques can be used to check the accuracy of fraction addition?
5. How does finding the least common multiple help in fraction operations?

Tip

When working with nested fractions, simplify from the innermost set of parentheses outward for a clear solution path.