What is the product?
$\left(3 a^2 b^7\right)\left(5 a^3 b^8\right)$
The goal is to simplify the algebraic expression \((3 a^2 b^7) \cdot (5 a^3 b^8)\). This involves multiplying coefficients and variable parts separately, applying the product rule for powers with the same base, which states \(x^m \cdot x^n = x^{m+n}\).
To simplify the expression, we group the numerical constants and the variable components:
\[(3 a^2 b^7) \cdot (5 a^3 b^8) = (3 \cdot 5) \cdot (a^2 \cdot a^3) \cdot (b^7 \cdot b^8)\]
First, multiply the coefficients: \(3 \cdot 5 = 15\).
Next, apply the exponent rule to the \(a\) terms: \(a^2 \cdot a^3 = a^{2+3} = a^5\).
Finally, apply the exponent rule to the \(b\) terms: \(b^7 \cdot b^8 = b^{7+8} = b^{15}\).
Combining the results, we obtain:
\[15 a^5 b^{15}\]
Comparing this result to the provided options:
The expression matches option d. Thus, the correct choice is d.
In computer science and data storage, calculating the total size of files or memory can be compared to this simplification process. For instance, if you have 3 files, each of size \(a^2 b^7\) megabytes, and you need to multiply this by 5 files of size \(a^3 b^8\), understanding how to combine these exponents allows for quick calculations of total data usage. Mastery of these algebraic rules is essential for managing variables in programming and engineering applications where complex data sets are multiplied frequently.