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10. \( (8-\sqrt{-16})(3+\sqrt{-81}) \)

Ask by Black Hall. in the United States
Feb 20,2025

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The result is \( 60 + 60i \).

Solución

Calculate or simplify the expression \( (8-\sqrt{-16})(3+\sqrt{-81}) \). Calculate the value by following steps: - step0: Calculate: \(\left(8-\sqrt{-16}\right)\left(3+\sqrt{-81}\right)\) - step1: Simplify the root: \(\left(8-4i\right)\left(3+\sqrt{-81}\right)\) - step2: Simplify the root: \(\left(8-4i\right)\left(3+9i\right)\) - step3: Apply the distributive property: \(8\times 3+8\times 9i-4i\times 3-4i\times 9i\) - step4: Multiply the numbers: \(24+8\times 9i-4i\times 3-4i\times 9i\) - step5: Multiply the numbers: \(24+72i-4i\times 3-4i\times 9i\) - step6: Multiply the numbers: \(24+72i-12i-4i\times 9i\) - step7: Multiply the numbers: \(24+72i-12i+36\) - step8: Calculate: \(60+72i-12i\) - step9: Multiply the numbers: \(60+60i\) The result of the expression \( (8-\sqrt{-16})(3+\sqrt{-81}) \) is \( 60+60i \).

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To solve the expression \( (8-\sqrt{-16})(3+\sqrt{-81}) \), we first simplify the square roots of the negative numbers. Starting with \(\sqrt{-16}\), we can rewrite it as: \[ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \] Now, for \(\sqrt{-81}\): \[ \sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i \] Now substituting back into the expression, we have: \[ (8 - 4i)(3 + 9i) \] Next, we can use the distributive property (also known as the FOIL method for binomials): \[ = 8 \cdot 3 + 8 \cdot 9i - 4i \cdot 3 - 4i \cdot 9i \] \[ = 24 + 72i - 12i - 36i^2 \] Here, recall that \(i^2 = -1\): \[ = 24 + 72i - 12i + 36 \] \[ = (24 + 36) + (72i - 12i) \] \[ = 60 + 60i \] Thus, the final simplified result of the expression \( (8-\sqrt{-16})(3+\sqrt{-81}) \) is: \[ \boxed{60 + 60i} \]

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