1Cademy - Multiplying $$(6u^2 - 11v^5)(6u^2 + 11v^5)$$ Using the Product of Conjugates Pattern

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Product of Conjugates Pattern

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Multiplying $(6u^2 - 11v^5)(6u^2 + 11v^5)$ Using the Product of Conjugates Pattern

Apply the Product of Conjugates Pattern to multiply $(6u^2 - 11v^5)(6u^2 + 11v^5)$ . Both binomials share the same first term $6u^2$ and the same last term $11v^5$ , with one using subtraction and the other addition, confirming they form a conjugate pair. Use the formula $(a - b)(a + b) = a^2 - b^2$ , where $a = 6u^2$ and $b = 11v^5$ .

Step 1 — Square the first term, $6u^2$ : Apply the Product to a Power Property to square the coefficient and variable separately: $6^2 = 36$ . Then apply the Power Property to the variable: $(u^2)^2 = u^{2 \cdot 2} = u^4$ . So $(6u^2)^2 = 36u^4$ .

Step 2 — Square the last term, $11v^5$ : Again, square the coefficient and apply the Power Property: $11^2 = 121$ and $(v^5)^2 = v^{5 \cdot 2} = v^{10}$ . So $(11v^5)^2 = 121v^{10}$ .

Step 3 — Write the difference of squares:

$(6u^2 - 11v^5)(6u^2 + 11v^5) = 36u^4 - 121v^{10}$

The product is $36u^4 - 121v^{10}$ . This example extends the Product of Conjugates Pattern to a conjugate pair in which both terms are higher-degree monomials — $6u^2$ (degree 2) and $11v^5$ (degree 5). Squaring such terms requires two exponent rules working together: the Product to a Power Property separates the coefficient from the variable so each can be squared independently, and the Power Property multiplies the exponents when a power is raised to another power (e.g., $(u^2)^2 = u^4$ and $(v^5)^2 = v^{10}$ ). The resulting difference of squares has degree 10, much higher than the degree-2 results seen in earlier conjugate examples.

Updated 2026-04-21

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OpenStax Elementary Algebra

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