Question 484636
You were given to simplify using the product rule the multiplication of:
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{{{(7x^6y^7)(5x^2y) }}}
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The first thing you may want to do is to multiply the coefficients (the 7 times the 5) and you get as the product 35. This makes the problem become:
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{{{35*(x^6y^7)(x^2y)}}}
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Next you may multiply the x terms together. Since both of the "x" terms have exponents, you multiply them by adding their exponents making the result as follows:
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{{{35*(x^(6+2))y^7*y}}}
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After you add the 6 + 2 the problem simplifies to:
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{{{35*x^8*y^7*y}}}
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Finally, you can multiply the two "y" terms by adding their exponents. Remember that the term {{{y}}} is the same as {{{y^1}}}. Therefore in adding the exponents of the "y" terms you get:
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{{{35*x^8*y^(7+1)}}}
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Adding the "y" exponents together results in the answer of:
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{{{35*x^8*y^8}}}
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Since both the "x" and the "y" terms have the same exponent you can also express this answer as:
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{{{35*(xy)^8}}}
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This last form of the answer is an example of the power rule which basically says that if you have the product of factors inside parentheses and raise this product to a common power you can do so by raising each of the factors to the common power. (Only this answer is in reverse. We had factors raised to a common power of 8 and changed form by raising the product of the factors to the common power.)
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Hope this helps you to understand your problem and the product rule a little better.