1Cademy - Simplifying $$(y^3)^6(y^5)^4$$ and $$(-6x^4y^5)^2$$ by Applying Several Exponent Properties

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Power Property for Exponents

Example

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

Some expressions require more than one exponent property to simplify fully. Recognizing which properties to apply — and in what order — is the key skill.

ⓐ $(y^3)^6(y^5)^4 = y^{38}$ :

Apply the Power Property to each factor separately. Multiply the exponents in each power-of-a-power: $(y^3)^6 = y^{3 \cdot 6} = y^{18}$ and $(y^5)^4 = y^{5 \cdot 4} = y^{20}$ . The expression becomes $y^{18} \cdot y^{20}$ .
Apply the Product Property. Both factors now share the base $y$ , so add the exponents: $y^{18} \cdot y^{20} = y^{18+20} = y^{38}$ .

ⓑ $(-6x^4y^5)^2 = 36x^8y^{10}$ :

Apply the Product to a Power Property. The base is the product $-6x^4y^5$ , which has three factors: $-6$ , $x^4$ , and $y^5$ . Raise each factor to the second power: $(-6)^2(x^4)^2(y^5)^2$ .
Apply the Power Property to the variable factors. Multiply the exponents: $(x^4)^2 = x^{4 \cdot 2} = x^8$ and $(y^5)^2 = y^{5 \cdot 2} = y^{10}$ .
Simplify the numerical power. Compute $(-6)^2 = 36$ . The final result is $36x^8y^{10}$ .

Part (a) illustrates a Power-then-Product sequence, while part (b) illustrates a Product-to-a-Power-then-Power sequence. In both cases, reducing the expression to a single power for each base is achieved by strategically chaining the properties together.

Updated 2026-04-21

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OpenStax Elementary Algebra

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