# Sums and Products with Exponents

Powers and roots can be distributed over products and quotients.  They may not be distributed over sums or differences, no matter how tempting it may be. Sums or differences raised to a power must be used as a factor the indicated number of times, then multiplied using the distributive property.

An exponent applies only to the factor immediately below it unless parentheses have been used to indicate otherwise.

“Like terms” have the same variables, to the same powers… and only “like” terms may be combined by adding their coefficients.

The answer shown below is wrong.  Try working your way through the problem backwards, from the answer up. As you find each mistake, try to identify the thinking behind it… what perspective led to the mistake? What should have been done instead?

$(f-g^2)^2-fg^2-(-fg)^2-f^2$

$=f^2-g^4-fg^2-(-fg)^2-f^2$

$=f^2-g^4-f^2g^2-(-fg)^2-f^2$

$=f^2-g^4-f^2g^2+fg^2-f^2$

$=-g^4$

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Hint 1: The correct answer to the problem on the first line above is: $g^4-3fg^2-f^2g^2$

Hint 2: The work shown above contains sign errors, simplification errors, and exponentiation errors. Re-read the text at the top of the posting if you have not found them all…