12g^(3)v^(4) * 4g^(2)v^(7)

Identify Equation and Apply Distributive Property: Identify the equation and apply the distributive property of exponents. We have two terms being multiplied, each with coefficients and variables raised to powers. We can combine like terms by adding the exponents of the same base and multiplying the coefficients.Multiply Coefficients of Terms: Multiply the coefficients (numerical parts) of the terms. The coefficients are 12 and 4. Multiplying these together gives us: 12 * 4 = 48Add Exponents of Variable g: Add the exponents of the variable g. The exponents on g are 3 and 2. When multiplying with the same base, we add the exponents: g^(3+2) = g^5Add Exponents of Variable v: Add the exponents of the variable v. The exponents on v are 4 and 7. When multiplying with the same base, we add the exponents: v^(4+7) = v^11Combine Results for Final Expression: Combine the results from steps 2, 3, and 4 to write the final simplified expression. We have the coefficient 48, g raised to the 5th power, and v raised to the 11th power. The simplified expression is: 48g^5v^11

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12 g^{3} v^{4} \times 4 g^{2} v^{7}

Identify Equation and Apply Distributive Property: Identify the equation and apply the distributive property of exponents. $\newline$ We have two terms being multiplied, each with coefficients and variables raised to powers. We can combine like terms by adding the exponents of the same base and multiplying the coefficients.
Multiply Coefficients of Terms: Multiply the coefficients (numerical parts) of the terms. $\newline$ The coefficients are $12$ and $4$ . Multiplying these together gives us: $\newline$ $12 \times 4 = 48$
Add Exponents of Variable $g$ : Add the exponents of the variable $g$ . The exponents on $g$ are $3$ and $2$ . When multiplying with the same base, we add the exponents: $g^{(3+2)} = g^5$
Add Exponents of Variable $v$ : Add the exponents of the variable $v$ . The exponents on $v$ are $4$ and $7$ . When multiplying with the same base, we add the exponents: $v^{4+7} = v^{11}$
Combine Results for Final Expression: Combine the results from steps $2$ , $3$ , and $4$ to write the final simplified expression. $\newline$ We have the coefficient $48$ , $g$ raised to the $5$ th power, and $v$ raised to the $11$ th power. The simplified expression is: $\newline$ $48g^5v^{11}$