Negative exponents represent the reciprocal or inverse of a positive exponent. They are used to express values that are fractions or decimals, rather than whole numbers, in the context of exponential expressions.
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Negative exponents can be rewritten as positive exponents in the denominator of a fraction.
The value of a negative exponent is the reciprocal of the positive exponent, so $x^{-n} = \frac{1}{x^n}$.
Negative exponents are useful when dividing monomials, as they can simplify the expression by moving the variable to the denominator.
When multiplying or dividing expressions with negative exponents, the exponents are added or subtracted, respectively.
Negative exponents can be used to represent very small values, such as $10^{-6}$ which is equivalent to 0.000001.
Review Questions
Explain how negative exponents can be used to simplify the division of monomials.
When dividing monomials, negative exponents can be used to move the variable to the denominator, simplifying the expression. For example, $\frac{x^4}{x^2} = x^{4-2} = x^2$. The negative exponent $-2$ in the denominator is equivalent to moving the $x^2$ term to the bottom of the fraction, resulting in $x^2$.
Describe the relationship between positive and negative exponents, and how they can be used to represent reciprocal values.
Negative exponents represent the reciprocal or inverse of a positive exponent. The value of $x^{-n}$ is equal to $\frac{1}{x^n}$. This relationship allows negative exponents to be used to express fractional or decimal values in exponential expressions. For instance, $5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04$.
Analyze how the rules for multiplying and dividing expressions with negative exponents differ from those with positive exponents.
When multiplying expressions with negative exponents, the exponents are added together, just as with positive exponents. However, when dividing expressions with negative exponents, the exponents are subtracted. For example, $x^{-3} \cdot x^{-2} = x^{-3-(-2)} = x^{-1} = \frac{1}{x}$, but $x^{-3} \div x^{-2} = x^{-3-(-2)} = x^{-1} = \frac{1}{x}$. This difference in the rules for multiplication and division with negative exponents is an important concept to understand when working with exponential expressions.