Nth Root

Definition

The nth root of a number is the value that, when raised to the power of n, equals the original number. It is a way of representing the inverse operation of exponentiation, where the exponent is a rational number rather than a whole number.

5 Must Know Facts For Your Next Test

1. The nth root of a number can be expressed using a rational exponent, where the numerator is the power and the denominator is the root.
2. The square root (2nd root) is a special case of the nth root, where n = 2.
3. The cube root (3rd root) is another common form of the nth root, where n = 3.
4. The nth root can be used to simplify and evaluate expressions involving rational exponents.
5. The properties of exponents, such as the power rule and the product rule, can be applied to expressions with rational exponents.

Review Questions

• Explain how the nth root is related to rational exponents.
  • The nth root of a number can be expressed using a rational exponent, where the numerator is the power and the denominator is the root. For example, the square root of 16 can be written as 16^(1/2), and the cube root of 27 can be written as 27^(1/3). This relationship allows us to apply the properties of exponents, such as the power rule and the product rule, to expressions involving rational exponents.
• Describe the process of evaluating an expression with a rational exponent.
  • To evaluate an expression with a rational exponent, such as $x^{2/3}$, we can first rewrite it as the cube root of $x^2$. This is because the denominator of the exponent (3) represents the root, and the numerator (2) represents the power. Then, we can apply the properties of exponents to simplify the expression further, if necessary. For example, $(x^2)^{1/3} = x^{2/3}$.
• Analyze how the properties of exponents can be used to manipulate expressions with rational exponents.
  • The properties of 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 applied to expressions with rational exponents. This allows us to simplify and evaluate these expressions more efficiently. For instance, if we have the expression $x^{2/3} \cdot x^{1/3}$, we can use the power rule to combine the exponents and simplify it to $x^{3/3} = x$. Understanding how to manipulate rational exponents using exponent properties is crucial for solving problems involving nth roots.

Related terms

Exponent: A number that indicates how many times the base number is to be multiplied by itself.

Radical: The symbol (√) used to represent the operation of finding the root of a number.

Rational Exponent: An exponent that can be expressed as a fraction, where the numerator is the power and the denominator is the root.