Rational Exponents

Definition

Rational exponents are a way to express roots using fractional powers. They allow you to rewrite expressions that involve roots in a form that is often easier to manipulate and calculate. This concept connects deeply with the laws of exponents, making it possible to work with both roots and powers in a unified manner, and provides a powerful tool for simplifying expressions and solving equations.

5 Must Know Facts For Your Next Test

1. The expression $$a^{m/n}$$ represents the nth root of $$a$$ raised to the mth power, allowing for a seamless connection between roots and exponents.
2. Rational exponents simplify calculations by allowing for the application of exponent laws, making it easier to multiply and divide terms with different bases and exponents.
3. A positive rational exponent indicates both a power and a root, which can lead to clearer solutions in equations involving polynomials or radical expressions.
4. When dealing with rational exponents, it's important to remember that the denominator of the fraction indicates the root, while the numerator indicates the power.
5. Expressions with rational exponents can be converted back into radical form for clarity, where $$a^{1/2}$$ becomes $$\sqrt{a}$$ and $$a^{1/3}$$ becomes $$\sqrt[3]{a}$$.

Review Questions

• How can you simplify an expression with rational exponents like $$x^{3/4} \cdot x^{1/2}$$?
  • To simplify $$x^{3/4} \cdot x^{1/2}$$, you apply the laws of exponents which state that when multiplying like bases, you add the exponents. First, convert $$1/2$$ to a fraction with a common denominator: $$x^{1/2} = x^{2/4}$$. Then add the exponents: $$3/4 + 2/4 = 5/4$$. Thus, the simplified expression is $$x^{5/4}$$.
• Explain how rational exponents can be beneficial in solving equations involving roots and powers.
  • Rational exponents streamline the process of solving equations that involve both roots and powers by providing a consistent way to handle these operations. Instead of dealing separately with radicals and polynomial expressions, rational exponents allow for all parts of an equation to be expressed in terms of exponentiation. This uniformity makes it easier to apply algebraic techniques such as factoring and simplifying expressions, ultimately leading to clearer solutions.
• Evaluate the expression $$16^{3/4}$$ and explain your reasoning process.
  • $$16^{3/4}$$ can be evaluated by first recognizing that this represents both a root and an exponent. Start by rewriting it as $$(16^{1/4})^3$$. The term $$16^{1/4}$$ is the fourth root of 16, which equals 2 (since $$2^4 = 16$$). Now raise this result to the power of 3: $$2^3 = 8$$. Therefore, $$16^{3/4} = 8$$, demonstrating how rational exponents simplify calculations involving roots.

Related terms

Radical: A radical is a symbol used to denote the root of a number, such as the square root or cube root.

Power: A power refers to the expression of a number raised to an exponent, indicating how many times the number is multiplied by itself.

Laws of Exponents: The laws of exponents are rules that describe how to handle mathematical operations involving exponents, including multiplication, division, and raising powers to powers.