D. Examples
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Example 1 |
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Simplify «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»72«/mn»«/msqrt»«/math».
Method 1: Using Prior Knowledge of Perfect Squares Step 1: Consider the factors of 72 (recall the Perfect Square chart you completed in the Check Up). 72 × 1, 36 × 2, 24 × 3, 18 × 4, 12 × 6, 9 × 8 Step 2: Identify the set of factors with the largest perfect square.
36 × 2 = 72
Step 3: Express «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»72«/mn»«/msqrt»«/math» as a product of its factors.
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt mathcolor=¨#B94A48¨»«mn»72«/mn»«/msqrt»«mo mathcolor=¨#B94A48¨»=«/mo»«msqrt mathcolor=¨#B94A48¨»«mn»36«/mn»«mo»§#215;«/mo»«mn»2«/mn»«/msqrt»«/math»
Step 4: Separate and give each factor its own root sign. Step 5: The «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt mathcolor=¨#0F1934¨»«msup»«mn»6«/mn»«mn»2«/mn»«/msup»«/msqrt»«mo mathcolor=¨#0F1934¨»=«/mo»«mn mathcolor=¨#0F1934¨»6«/mn»«/math», so rewrite the entire radical as a simplified mixed radical. «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt mathcolor=¨#B94A48¨»«mn»72«/mn»«/msqrt»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»6«/mn»«msqrt mathcolor=¨#B94A48¨»«mn»2«/mn»«/msqrt»«/math» |
Another method of simplifying radicals involves using prime factorization.
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Example 2 |
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Simplify «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»72«/mn»«/msqrt»«/math».
Method 2: Using Prior Knowledge of Prime Factorization Step 1: Using a prime factorization tree, find the prime factors of 72.
Step 2: The index of «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»72«/mn»«/msqrt»«/math» is implied to be 2 because «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»72«/mn»«/msqrt»«/math» is a square root. As such, identify the prime factors of 72 that repeat (doubles).
Step 3: Rewrite the radical as a product of its prime factors.
Step 4: Rewrite the entire radical as a simplified mixed radical.
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