1Cademy - Multiplying $$(cd - 8)(cd + 8)$$ Using the Product of Conjugates Pattern

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Product of Conjugates Pattern

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Multiplying $(cd - 8)(cd + 8)$ Using the Product of Conjugates Pattern

Apply the Product of Conjugates Pattern to multiply $(cd - 8)(cd + 8)$ . Both binomials share the same first term $cd$ and the same last term $8$ , with one using subtraction and the other addition, so they form a conjugate pair. Use the formula $(a - b)(a + b) = a^2 - b^2$ , where $a = cd$ and $b = 8$ .

Step 1 — Square the first term, $cd$ : $(cd)^2 = c^2d^2$ . When a product of two variables is squared, the exponent applies to each variable individually: $c^2$ and $d^2$ .

Step 2 — Square the last term, $8$ : $8^2 = 64$ .

Step 3 — Write the difference of squares:

$(cd - 8)(cd + 8) = c^2d^2 - 64$

The product is $c^2d^2 - 64$ . This example illustrates the pattern when the variable term is itself a product of two variables ( $cd$ ). Squaring $cd$ requires distributing the exponent to each variable factor using the Product to a Power Property: $(cd)^2 = c^2 \cdot d^2 = c^2d^2$ , not $cd^2$ . Students must recognize that the entire expression $cd$ serves as the "first term" in the conjugate pair.

Updated 2026-04-21

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OpenStax Elementary Algebra

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