Sure, let's prove the distributive property of multiplication over addition for the numbers [tex]\(2\)[/tex], [tex]\(\frac{4}{15}\)[/tex], and [tex]\(\frac{7}{15}\)[/tex].
The distributive property of multiplication over addition states that for any three real numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex],
[tex]\[ a \cdot (b + c) = (a \cdot b) + (a \cdot c) \][/tex]
Let's assign the values as follows:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = \frac{4}{15} \][/tex]
[tex]\[ c = \frac{7}{15} \][/tex]
We need to verify that:
[tex]\[ 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) = \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) \][/tex]
First, let's calculate the left side of the equation:
### Left Side: [tex]\( a \cdot (b + c) \)[/tex]
[tex]\[ a \cdot (b + c) = 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) \][/tex]
Combine the fractions inside the parentheses:
[tex]\[ \frac{4}{15} + \frac{7}{15} = \frac{4 + 7}{15} = \frac{11}{15} \][/tex]
Now multiply by [tex]\( a = 2 \)[/tex]:
[tex]\[ 2 \cdot \frac{11}{15} = \frac{2 \cdot 11}{15} = \frac{22}{15} \approx 1.4666666666666668 \][/tex]
### Right Side: [tex]\((a \cdot b) + (a \cdot c)\)[/tex]
Calculate each term individually:
[tex]\[ a \cdot b = 2 \cdot \frac{4}{15} = \frac{2 \cdot 4}{15} = \frac{8}{15} \approx 0.5333333333333333 \][/tex]
[tex]\[ a \cdot c = 2 \cdot \frac{7}{15} = \frac{2 \cdot 7}{15} = \frac{14}{15} \approx 0.9333333333333333 \][/tex]
Add these results together:
[tex]\[ \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) = \frac{8}{15} + \frac{14}{15} = \frac{8 + 14}{15} = \frac{22}{15} \approx 1.4666666666666668 \][/tex]
### Conclusion
Both the left side and the right side of the equation simplify to [tex]\(\frac{22}{15}\)[/tex], and their approximate decimal values are [tex]\(1.4666666666666668\)[/tex].
Therefore, the distributive property holds true in this case:
[tex]\[ 2 \cdot \left( \frac{4}{15} + \frac{7}{15} \right) = \left( 2 \cdot \frac{4}{15} \right) + \left( 2 \cdot \frac{7}{15} \right) \][/tex]
So, we have proven that the distributive property of multiplication over addition is valid for the given numbers.
