SOLUTION: Hi, my teacher provided problems and answers and asked us to find out how one gets to the other. Example: {{{ 5 sqrt ( 18 ) }}} equals {{{ 15 sqrt ( 2 ) }}}. I tried simplifying

Algebra ->  Square-cubic-other-roots -> SOLUTION: Hi, my teacher provided problems and answers and asked us to find out how one gets to the other. Example: {{{ 5 sqrt ( 18 ) }}} equals {{{ 15 sqrt ( 2 ) }}}. I tried simplifying       Log On


   

Question 531134: Hi, my teacher provided problems and answers and asked us to find out how one gets to the other. Example: +5+sqrt+%28+18+%29+ equals +15+sqrt+%28+2+%29+. I tried simplifying it by bringing it back to its original state root 42, but could not come up with an answer because there were no prime number pairs when I was factoring. I am stumped; thank you so much in advance for your help.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given to simplify:
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5%2Asqrt%2818%29
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Replace 18 by its equivalent 9 times 2 and you have:
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5%2Asqrt%289%2A2%29
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But the square root of this product is equal to the product of the square roots. This becomes:
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5%2Asqrt%289%2A2%29+=+5%2Asqrt%289%29%2Asqrt%282%29
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But the square root of 9 is 3. Replace the square root of 9 with 3 and the term is:
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5%2A3%2Asqrt%282%29
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Multiply the 5 and the 3 and the result is:
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15%2Asqrt%282%29
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The critical part comes from the rule:
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sqrt%28a%2Ab%29+=+sqrt%28a%29%2Asqrt%28b%29
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For this problem you can verify this rule by using a calculator to find the square root of 18 (which is equal to the square root of (9*2). Then find the square root of 9 (answer 3) and multiply it by the square root of 2. This product should equal the square root of 18. You will find that it does. This does not prove the rule, but it should give you some confidence that the rule does work.
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Hope this helps you with the problem.
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