Common Types of Mistakes: Problems with Exponents
The rules of exponents are often used incorrectly. Here are some examples.
Example 1: (Multiplication)
Mistake:
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Multiply two (or more) factors that have the same base by adding (not multiplying) the exponents.
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Click to see an explanation. Click to hide the explanation.
The explanation shows why the rule says to add the exponents, at least for positive integer powers. The rule applies to all powers, including negative and fractional powers.
Test your understanding by simplifying the following expression and then check your answer:
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Example 2: (Division)
Mistake:
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Divide factors that are the same base by subtracting (not dividing) the exponents.
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Click to see an explanation. Click to hide the explanation.
The explanation shows why the rule says to subtract the exponents, at least for positive integer powers. The rule applies to all powers, including negative and fractional powers.
Test your understanding by simplifying the following expression and then check your answer:
Click to check your answer. Click to remove the answer.
Example 3: (Exponent)
Mistake:
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Raise an exponent expression to another exponent by multiplying the exponents.
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Click to see an explanation. Click to hide the explanation.
The explanation shows why the rule says to multiply the exponents, at least for positive integer powers. The rule applies to all powers, including negative and fractional powers.
Test your understanding by simplifying the following expression and then check your answer:
Click to check your answer. Click to remove the answer.
Example 4: (Negative Exponent)
Mistake:
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A negative exponent denotes a reciprocal expression.
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The explanation, which uses the rule mentioned in Example 2 for dividing exponent expressions with a common base and the fact that x0 = 1, shows why a negative exponent denotes a reciprocal. The reciprocal meaning applies to all negative exponents, including negative fractions.
Example 5: (Fractional Exponent)
Mistake:
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A negative exponent denotes a reciprocal expression.
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The explanation shows why a fractional exponent denotes a radical. Raising the expression with a fraction as the exponent to the power equal to the denominator of the exponent is accomplished by multiplying the exponents, thus eliminating the fractional exponent.
Example 6: (Precedence of Exponents)
Mistake:
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Taking a power has a higher precedence than multiplication, so the power -1 only applies to x, not to all of 3x.
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If you intend that all of 3x is raised to the power -1, then use parentheses: (3x)-1.
Eliminate mistakes with exponents by learning and using the rules of exponents correctly.
