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 represents the inverse operation of exponentiation, where the nth root extracts the value that was raised to the power of n.

5 Must Know Facts For Your Next Test

1. The nth root of a number can be used to simplify rational exponents, where the denominator of the exponent represents the root index.
2. When dividing radical expressions, the nth roots can be used to rationalize the denominator by applying the property of nth roots.
3. Radicals, which represent nth roots, can be used in functions to model real-world phenomena, such as the relationship between the area of a circle and its radius.
4. The value of the nth root can be found by raising the base to the power of 1/n, which is the inverse of the original exponent.
5. The properties of nth roots, such as the product rule and the power rule, can be used to simplify and manipulate radical expressions.

Review Questions

• Explain how the concept of nth roots is used to simplify rational exponents in the context of Section 8.3.
  • In Section 8.3, which covers simplifying rational exponents, the concept of nth roots is used to convert the rational exponent into a radical expression. The denominator of the rational exponent represents the root index, so the numerator of the exponent can be used as the power to which the base is raised. For example, $x^{3/4}$ can be rewritten as $\sqrt[4]{x^3}$, where the 4th root is taken of $x^3$. This allows for the simplification of rational exponents by converting them into radical form.
• Describe how the properties of nth roots can be used to divide radical expressions in the context of Section 8.5.
  • In Section 8.5, which focuses on dividing radical expressions, the properties of nth roots can be utilized to rationalize the denominator. By applying the property that $\sqrt[n]{a} / \sqrt[n]{b} = \sqrt[n]{a/b}$, the denominator can be simplified by taking the nth root of the numerator and denominator separately. This process of rationalizing the denominator using nth roots is important for simplifying and evaluating divided radical expressions.
• Discuss how radicals, which represent nth roots, can be used in functions to model real-world phenomena in the context of Section 8.7.
  • Section 8.7 explores the use of radicals in functions, where the nth root can be used to model various real-world relationships. For example, the area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius. This can be rewritten as $A = \pi \sqrt[2]{r^2}$, which is the square root of the radius squared. Similarly, the volume of a sphere is given by $V = \frac{4}{3}\pi r^3$, which can be expressed as $V = \frac{4}{3}\pi \sqrt[3]{r^3}$, the cube root of the radius cubed. These examples demonstrate how radicals, representing nth roots, can be used in functions to model and understand real-world phenomena.

Related terms

Radical: A radical is the symbol used to represent the nth root of a number, typically denoted by the $\sqrt[n]{}$ symbol.

Exponent: An exponent is the power to which a number or variable is raised, and the nth root is the inverse operation that extracts the value of the base that was raised to the power of n.

Rational Exponent: A rational exponent is an exponent that can be expressed as a fraction, and it is related to nth roots as the denominator of the fraction represents the root index.