# 4 Common Errors in Calculating Expressions with Exponents

The errors below are usually made by students in calculating algebraic expressions with integer exponents. These errors result from the misunderstanding of the law of exponents.

If you have seen  errors like the following, please don’t hesitate to use the comment box below.

Common Error 1$-a^n = (-a)^n$

That equation is only true if $n$ is odd. If $n$ is even, the equation does not hold. For example, in the expression $-3^4$, the exponent $4$ only applies to $3$ and not $-3$. That means that $-3^4 = - 81$. However, $(-3)^4 = 81$

Common Error 2: $ab^n = a^nb^n$

This error usually results from the law of exponent that $(ab)^n = a^nb^n$. Note that in the absence of parentheses and brackets, the exponent only applies to the variable to which it is attached. So, in $ab^n$, only $b$ is raised to $n$. Therefore, $ab^n$ is not equal to $a^nb^n$.

Common Error 3: $(a + b)^n = a^n + b^n$.

This is only true if $n = 1$. The most common error are $(a + b)^2 = a^2 + b^2$ and $(a + b)^3 = a^3 + b^3$. Of course, we need to emphasize that $(a + b)^2 = (a+b)(a+b)$ and $(a+b)^3 = (a+b)(a+b)(a+b)$.

Common Error 4:  $a^m\cdot a^n=a^m+a^n$

The expression $a^m\cdot a^n=a^{m+n}$, but the equation above does not hold. For example, $2^3\cdot 2^4=2^3 \cdot 2^4 = 2^7 = 128$, while $2^3 + 2^4 = 8 + 16 = 24$.

Reference: Explorations in College Algebra, Linda Kime, et al.