So seeing a classic mathematical mistake in a recent thread prompted me to list out 5 of the most common mistakes I see kids make. Feel free to add to my list as I know there are many more out there. I figure this thread can also do a lot of good by consolidating many of the mistakes that we see posters on this forum do and thus, hopefully, not continue to do.
1) INCORRECT: Improper expansion of a binomial:\(\displaystyle (x\pm 3)^2 = x^2+9\)
CORRECT: \(\displaystyle (x\pm 3)^2 = x^2\pm 6x+9\)
2) INCORRECT: Taking the square root of each individual term:\(\displaystyle \sqrt{x^2+9} = x+3\)
CORRECT: \(\displaystyle \sqrt{x^2+9}=\ literally \ \sqrt{x^2+9}\) Obviously, if \(\displaystyle x^2\) was an actual number than just add it to 9 and take the square root of the resulting number.
3) INCORRECT: Cancelling a term when it is being added or subtracted with another term:\(\displaystyle \frac{x+5}{x}=\frac{\otimes+5}{\otimes}=5\) (The \(\displaystyle \otimes\) means the "x" is being cancelled. Couldn't figure out how to do a strikethrough on it
)
CORRECT: \(\displaystyle \frac{x+5}{x}=\frac{x}{x}+\frac{5}{x}=1+\frac{5}{x}\) (for this specific example)
4) INCORRECT: Failing to apply the exponent to a number when simplifying a rational expression:\(\displaystyle (\frac{4x^4y^2}{z^3})^3 = \frac{4x^{12}y^6}{z^9}\)
CORRECT: \(\displaystyle (\frac{4x^4y^2}{z^3})^3 = \frac{4^3x^{12}y^6}{z^9}=\frac{64x^{12}y^6}{z^9}\)
5) INCORRECT: Thinking a number raised to a -1 makes the value negative:\(\displaystyle 5^{-2} = -25\)
CORRECT: \(\displaystyle 5^{-2} = \frac{1}{5^2}=\frac{1}{25}\)
1) INCORRECT: Improper expansion of a binomial:\(\displaystyle (x\pm 3)^2 = x^2+9\)
CORRECT: \(\displaystyle (x\pm 3)^2 = x^2\pm 6x+9\)
2) INCORRECT: Taking the square root of each individual term:\(\displaystyle \sqrt{x^2+9} = x+3\)
CORRECT: \(\displaystyle \sqrt{x^2+9}=\ literally \ \sqrt{x^2+9}\) Obviously, if \(\displaystyle x^2\) was an actual number than just add it to 9 and take the square root of the resulting number.
3) INCORRECT: Cancelling a term when it is being added or subtracted with another term:\(\displaystyle \frac{x+5}{x}=\frac{\otimes+5}{\otimes}=5\) (The \(\displaystyle \otimes\) means the "x" is being cancelled. Couldn't figure out how to do a strikethrough on it
)CORRECT: \(\displaystyle \frac{x+5}{x}=\frac{x}{x}+\frac{5}{x}=1+\frac{5}{x}\) (for this specific example)
4) INCORRECT: Failing to apply the exponent to a number when simplifying a rational expression:\(\displaystyle (\frac{4x^4y^2}{z^3})^3 = \frac{4x^{12}y^6}{z^9}\)
CORRECT: \(\displaystyle (\frac{4x^4y^2}{z^3})^3 = \frac{4^3x^{12}y^6}{z^9}=\frac{64x^{12}y^6}{z^9}\)
5) INCORRECT: Thinking a number raised to a -1 makes the value negative:\(\displaystyle 5^{-2} = -25\)
CORRECT: \(\displaystyle 5^{-2} = \frac{1}{5^2}=\frac{1}{25}\)
