1Cademy - Simplifying $$(5m)^2(3m^3)$$ and $$(3x^2y)^4(2xy^2)^3$$ Using Several Exponent Properties

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Simplifying $(y^3)^6(y^5)^4$ and $(-6x^4y^5)^2$ by Applying Several Exponent Properties

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Simplifying $(5m)^2(3m^3)$ and $(3x^2y)^4(2xy^2)^3$ Using Several Exponent Properties

When two products-raised-to-powers are multiplied together, simplify by applying the Product to a Power Property to each factor first, then using the Commutative Property to group the numerical coefficients and like variable bases together, and finally multiplying the constants and adding the exponents for each base.

ⓐ $(5m)^2(3m^3) = 75m^5$ :

Apply the Product to a Power Property to $(5m)^2$ . Raise each factor inside the parentheses to the second power: $(5m)^2 = 5^2 \cdot m^2$ . Simplify: $5^2 = 25$ . The expression becomes $25m^2 \cdot 3m^3$ .
Use the Commutative Property to rearrange the factors so that constants are together and like bases are together: $25 \cdot 3 \cdot m^2 \cdot m^3$ .
Multiply the constants and add the exponents: $25 \cdot 3 = 75$ and $m^2 \cdot m^3 = m^{2+3} = m^5$ . The result is $75m^5$ .

ⓑ $(3x^2y)^4(2xy^2)^3 = 648x^{11}y^{10}$ :

Apply the Product to a Power Property to each factor. For the first: $(3x^2y)^4 = 3^4 \cdot (x^2)^4 \cdot y^4 = 81x^8y^4$ . For the second: $(2xy^2)^3 = 2^3 \cdot x^3 \cdot (y^2)^3 = 8x^3y^6$ . The expression becomes $(81x^8y^4)(8x^3y^6)$ .
Use the Commutative Property to group constants and like bases: $81 \cdot 8 \cdot x^8 \cdot x^3 \cdot y^4 \cdot y^6$ .
Multiply the constants and add the exponents for each base: $81 \cdot 8 = 648$ , $x^8 \cdot x^3 = x^{11}$ , and $y^4 \cdot y^6 = y^{10}$ . The result is $648x^{11}y^{10}$ .

Part (b) extends the technique to expressions involving multiple variables, requiring the Product Property to be applied separately to each variable base. In both parts, the Commutative Property is the key organizational step that makes the final multiplication and exponent addition straightforward.

Updated 2026-04-29

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