Pregunta
19. Expand \( (8-\sqrt{3})^{2} \) giving your answer in form \( a+b \sqrt{3} \)
Ask by Bowen French. in Thailand
Mar 14,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
\( 67 - 16\sqrt{3} \)
Solución
1. Write the expression to be expanded:
\[
(8 - \sqrt{3})^2
\]
2. Use the formula for the square of a binomial:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
Here, \(a = 8\) and \(b = \sqrt{3}\).
3. Substitute the values into the formula:
\[
(8 - \sqrt{3})^2 = 8^2 - 2 \cdot 8 \cdot \sqrt{3} + (\sqrt{3})^2
\]
4. Compute each term:
\[
8^2 = 64
\]
\[
2 \cdot 8 \cdot \sqrt{3} = 16\sqrt{3}
\]
\[
(\sqrt{3})^2 = 3
\]
5. Combine the results:
\[
64 - 16\sqrt{3} + 3 = 67 - 16\sqrt{3}
\]
6. Therefore, the expanded form is:
\[
67 - 16\sqrt{3}
\]
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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The Deep Dive
To expand \((8-\sqrt{3})^{2}\), we use the formula \((a-b)^{2} = a^{2} - 2ab + b^{2}\). Here, \(a = 8\) and \(b = \sqrt{3}\). Calculating each term: 1. \(a^{2} = 8^{2} = 64\) 2. \(2ab = 2 \cdot 8 \cdot \sqrt{3} = 16\sqrt{3}\) 3. \(b^{2} = (\sqrt{3})^{2} = 3\) Plugging these values into the formula, we get: \[ (8-\sqrt{3})^{2} = 64 - 16\sqrt{3} + 3 = 67 - 16\sqrt{3} \] So, the final answer in the form \(a + b\sqrt{3}\) is \(\boxed{67 - 16\sqrt{3}}\).

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