We are given the algebraic expression (4x squared y) times (3x cubed y cubed). Our goal is to simplify this product of two monomials. Let's break it down systematically.
Let's separate the expression into its components. We have coefficients 4 and 3, x terms x squared and x cubed, and y terms y and y cubed. We multiply coefficients separately from variables.
First, we multiply the coefficients 4 and 3. Four times three equals twelve. Now our expression becomes 12 times x squared y times x cubed y cubed.
To multiply variables with the same base, we add their exponents. The rule is x to the power of a times x to the power of b equals x to the power of a plus b. For example, x squared times x cubed equals x to the fifth power.
Now let's multiply the x terms: x squared times x cubed. Using our exponent rule, we add the exponents: 2 plus 3 equals 5. So x squared times x cubed equals x to the fifth power. Our expression is now 12x to the fifth power times y times y cubed.
Finally, we multiply the y terms. Remember that y is the same as y to the first power. So we have y to the first times y cubed. Adding the exponents: 1 plus 3 equals 4. Therefore, y times y cubed equals y to the fourth power.
Let's summarize our simplification process. Starting with (4x squared y) times (3x cubed y cubed), we multiplied coefficients: 4 times 3 equals 12. We multiplied x terms: x squared times x cubed equals x to the fifth. We multiplied y terms: y times y cubed equals y to the fourth. The final simplified form is 12x to the fifth y to the fourth.
Let's practice with another example: (2a cubed b squared) times (5a squared b to the fourth). Multiply coefficients: 2 times 5 equals 10. Multiply a terms: a cubed times a squared equals a to the fifth. Multiply b terms: b squared times b to the fourth equals b to the sixth. The answer is 10a to the fifth b to the sixth.