Show that:
[tex]\[
\left(\frac{-8}{9} \times \frac{-1}{5}\right) + \left(\frac{-8}{9} \times \frac{-7}{11}\right) = \frac{-8}{9} \times \left(\frac{-1}{5} + \frac{-7}{11}\right)
\][/tex]



Answer :

Sure! Let's go through the steps for proving the given equation:

[tex]\[ \left(\frac{-8}{9} \times \frac{-1}{5}\right) + \left(\frac{-8}{9} \times \frac{-7}{11}\right) = \frac{-8}{9} \times \left(\frac{-1}{5} + \frac{-7}{11}\right) \][/tex]

### Step 1: Calculate the Left-Hand Side (LHS)

First, we will calculate each term on the left-hand side separately.

1. First Term: [tex]\(\frac{-8}{9} \times \frac{-1}{5}\)[/tex]

[tex]\[ \frac{-8}{9} \times \frac{-1}{5} = \frac{8}{45} \approx 0.17777777777777776 \][/tex]

2. Second Term: [tex]\(\frac{-8}{9} \times \frac{-7}{11}\)[/tex]

[tex]\[ \frac{-8}{9} \times \frac{-7}{11} = \frac{56}{99} \approx 0.5656565656565656 \][/tex]

Next, we add the two results to get the total left-hand side:

[tex]\[ \frac{8}{45} + \frac{56}{99} \approx 0.17777777777777776 + 0.5656565656565656 = 0.7434343434343433 \][/tex]

### Step 2: Calculate the Right-Hand Side (RHS)

Now, we will calculate the right-hand side of the equation:

1. Sum Inside the Parentheses: [tex]\(\frac{-1}{5} + \frac{-7}{11}\)[/tex]

First, get a common denominator (LCM of 5 and 11 is 55):

[tex]\[ \frac{-1}{5} = \frac{-11}{55}, \quad \frac{-7}{11} = \frac{-35}{55} \][/tex]

Now, add the fractions:

[tex]\[ \frac{-11}{55} + \frac{-35}{55} = \frac{-46}{55} \][/tex]

2. Multiply by [tex]\(\frac{-8}{9}\)[/tex]: [tex]\(\frac{-8}{9} \times \frac{-46}{55}\)[/tex]

[tex]\[ \frac{-8}{9} \times \frac{-46}{55} = \frac{368}{495} \approx 0.7434343434343434 \][/tex]

### Step 3: Comparison

Finally, compare the numerical values of the left-hand side and the right-hand side:

[tex]\[ 0.7434343434343433 \quad \text{and} \quad 0.7434343434343434 \][/tex]

We note a small numerical difference indicating that the exact equality does not hold perfectly in precise decimal arithmetic. Thus,

[tex]\[ \left(\frac{-8}{9} \times \frac{-1}{5}\right) + \left(\frac{-8}{9} \times \frac{-7}{11}\right) \ne \frac{-8}{9} \times \left(\frac{-1}{5} + \frac{-7}{11}\right) \][/tex]

The mathematical identities involved assume precise calculation, but due to splitting and combining floating-point arithmetic, there may appear discrepancies. Therefore, in this example, the given equality does not hold exactly.