Let's simplify each expression one by one:
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Simplifying \( \sqrt{108} \): \[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} \] Response: \( 6\sqrt{3} \)
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Simplifying \( \sqrt{15} \cdot 5 \): \[ \sqrt{15} \cdot 5 = 5\sqrt{15} \] Response: \( 5\sqrt{15} \)
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Simplifying \( \sqrt{1223} \): On finding the prime factors of 1223, we note that it is prime itself, hence: \[ \sqrt{1223} = \sqrt{1223} \] Response: \( \sqrt{1223} \)
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Simplifying \( \sqrt{43 + 27} \): \[ \sqrt{43 + 27} = \sqrt{70} \] Response: \( \sqrt{70} \)
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Simplifying \( \sqrt{312 - 23} \): \[ \sqrt{312 - 23} = \sqrt{289} = 17 \] Response: \( 17 \)
Now, let’s summarize the simplified results:
- \( \sqrt{108} = 6\sqrt{3} \)
- \( \sqrt{15} \cdot 5 = 5\sqrt{15} \)
- \( \sqrt{1223} = \sqrt{1223} \)
- \( \sqrt{43 + 27} = \sqrt{70} \)
- \( \sqrt{312 - 23} = 17 \)
These are the final simplest forms for the given expressions.