Rational Exponents

Definition

Rational exponents are a way of expressing fractional or negative exponents using a rational number, such as a fraction or a decimal. They provide a consistent way to represent and evaluate expressions involving powers with non-integer exponents.

5 Must Know Facts For Your Next Test

1. Rational exponents can be used to represent both fractional and negative powers in a consistent way.
2. The value of an expression with a rational exponent is equal to the base raised to the power of the numerator, and then taking the root specified by the denominator.
3. Rational exponents follow the same rules as integer exponents, such as the power rule and the product rule, allowing for simplification and manipulation of expressions.
4. Rational exponents are particularly useful in representing and working with quantities that involve roots, such as square roots, cube roots, and higher-order roots.
5. Understanding rational exponents is crucial for working with expressions involving fractional and negative powers, which are commonly encountered in various mathematical and scientific contexts.

Review Questions

• Explain how to evaluate an expression with a rational exponent, such as $x^{2/3}$.
  • To evaluate an expression with a rational exponent, such as $x^{2/3}$, you first raise the base ($x$) to the power of the numerator (2), and then take the root specified by the denominator (3). This means that $x^{2/3}$ is equivalent to the cube root of $x$ raised to the power of 2. In other words, $x^{2/3} = \sqrt[3]{x^2}$.
• Describe the relationship between fractional exponents and roots.
  • Fractional exponents and roots are closely related. A fractional exponent with a numerator of $n$ and a denominator of $m$ is equivalent to taking the $m$-th root of the base raised to the power of $n$. For example, $x^{2/3}$ is the same as the cube root of $x$ raised to the power of 2, or \sqrt[3]{x^2}. This connection allows for the simplification and manipulation of expressions involving roots using the properties of rational exponents.
• Analyze how the properties of integer exponents, such as the power rule and the product rule, can be applied to rational exponents.
  • The properties of integer exponents, such as the power rule ($x^a \cdot x^b = x^{a+b}$) and the product rule ($x^a \cdot y^a = (xy)^a$), can be extended to rational exponents. For example, the power rule for rational exponents states that $(x^{a/b})^{c/d} = x^{(a \cdot c) / (b \cdot d)}$, and the product rule states that $(x^{a/b}) \cdot (x^{c/d}) = x^{(a/b + c/d)}$. These properties allow for the simplification and manipulation of expressions involving rational exponents, which is crucial in various mathematical and scientific applications.

Related terms

Exponent: A number that indicates how many times the base is used as a factor in a multiplication expression.

Fractional Exponents: Exponents expressed as a fraction, such as $x^{2/3}$, which represents the cube root of $x$ raised to the power of 2.

Negative Exponents: Exponents with a negative value, such as $x^{-2}$, which represents the reciprocal of $x$ squared.