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Simplify. (x^2+8xy+16y^2)^1/3 times (x+4y)^1/3
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High School Teacher
To simplify, notice that `x^2 +8xy + 16y^2` is a perfect square trinomial and can be factored as:``
`x^2 +8xy +16y^2 = (x+4y)^2`
So, you can rewrite the expression as:
`((x+4y)^2)^(1/3) * (x+4y)^(1/3)`
Then, you can simplify it using the property of exponent:
`(a^n)^m = a^(mn)`
`n = 2`
So, `((x+4y)^2)^(1/3) = (x+4y)^(2/3)`
You now have: `(x+4y)^(2/3) * (x+4y)^(1/3)`
Notice that the two terms have the same base (x+4y). So you can use, `a^n * a^m = a^(m+n)`
`m = 1/3`
So, `(x+4y)^(2/3+1/3) = (x+4y)^1`
Thus, the answer is `x+4y`
You can check your answer by assuming values for x and y. Then, plug-in in the original expression and the final answer. They should give the same value.
Say, x = 2 and y =3.
`(x^2 +8xy +16y^2)^(1/3) * (x+4y)^(1/3)`
`2 + 4*3 = 14`
Posted by mariloucortez on April 16, 2013 at 1:45 AM (Answer #1)
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