Which expression is equivalent to $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ ?
The expression to simplify is \(\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2\). To simplify this, we need to apply standard rules of exponents. Specifically, we will use the power of a product rule \((ab)^n = a^n b^n\), the power of a power rule \((a^m)^n = a^{m \cdot n}\), and the product of powers rule \(a^m \cdot a^n = a^{m+n}\). Our objective is to find which of the given options matches our simplified result.
First, we evaluate the term \(\left(3 x^5 y\right)^2\) using the power of a product rule \((abc)^n = a^n b^n c^n\). Applying this to the expression, we get:
\[\left(3 x^5 y\right)^2 = 3^2 \cdot (x^5)^2 \cdot y^2\]
Calculating the powers, we have \(3^2 = 9\), and by the power of a power rule, \((x^5)^2 = x^{5 \cdot 2} = x^{10}\). Thus, the expression becomes:
\[9 x^{10} y^2\]
Now, we substitute this back into the original expression:
\[\left(4 x^3 y^5\right) \cdot \left(9 x^{10} y^2\right)\]
We can group the constants and the variables with the same bases together:
\[(4 \cdot 9) \cdot (x^3 \cdot x^{10}) \cdot (y^5 \cdot y^2)\]
Calculating the product of the constants, we have \(4 \cdot 9 = 36\). For the variables, we use the product of powers rule \(a^m \cdot a^n = a^{m+n}\):
\[x^3 \cdot x^{10} = x^{3+10} = x^{13}\]
\[y^5 \cdot y^2 = y^{5+2} = y^7\]
Combining these, we get:
\[36 x^{13} y^7\]
Comparing our result, \(36 x^{13} y^7\), with the given options:
We clearly see that our result matches option (b). The equivalent expression is \(\boxed{b}\).
In scientific computing and data modeling, complex expressions involving physical parameters (like velocity, density, or time) often need to be simplified before being computed. For example, if you are modeling energy in a system using units that result in the expression \((4v^3t^5)(3v^5t)^2\), simplifying this to \(36v^{13}t^7\) makes it much easier to perform calculations or input the formula into software for simulation. Mastering exponent rules ensures that you can manage such algebraic manipulations efficiently and accurately in any analytical field.