Negative exponents represent the reciprocal or inverse of the base number raised to a positive exponent. They are used to simplify expressions, divide monomials, and work with scientific notation and rational exponents.
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Negative exponents represent the reciprocal or inverse of the base number raised to a positive exponent.
When dividing monomials, negative exponents can be used to simplify the expression by moving the variable to the denominator.
In scientific notation, negative exponents are used to represent very small numbers as a decimal between 1 and 10 multiplied by a negative power of 10.
Rational exponents, such as $\frac{1}{2}$ or $-\frac{3}{4}$, can be rewritten using negative exponents to simplify calculations.
Negative exponents follow the same rules as positive exponents, but the exponent is treated as the opposite (negative) value.
Review Questions
Explain how negative exponents are used in the multiplication properties of exponents.
When multiplying expressions with negative exponents, the exponents are added together. For example, $x^{-2} \times x^{-3} = x^{-2 + (-3)} = x^{-5}$. This simplifies the expression by combining the negative exponents into a single negative exponent. The result represents the reciprocal of $x$ raised to the 5th power, which is $\frac{1}{x^5}$.
Describe how negative exponents are used when dividing monomials.
When dividing monomials, negative exponents can be used to move variables from the numerator to the denominator, simplifying the expression. For example, $\frac{x^3y^2}{x^5y} = x^{3-5}y^{2-1} = x^{-2}y^1 = \frac{1}{x^2}y$. The negative exponent on $x$ indicates that the variable is now in the denominator, representing the reciprocal of $x^2$.
Analyze how negative exponents are used in scientific notation and rational exponents.
In scientific notation, negative exponents are used to represent very small numbers as a decimal between 1 and 10 multiplied by a negative power of 10. For example, $0.000456$ can be written as $4.56 \times 10^{-4}$, where the negative exponent indicates that the decimal point should be moved 4 places to the left. Similarly, rational exponents like $\frac{1}{2}$ or $-\frac{3}{4}$ can be rewritten using negative exponents as $x^{-\frac{1}{2}}$ or $x^{-\frac{3}{4}}$, respectively, to simplify calculations involving roots and fractional powers.