Product Property of Roots

Definition

The product property of roots states that the nth root of a product is equal to the product of the nth roots of the individual factors. This property allows for simplification and evaluation of expressions involving roots of products.

5 Must Know Facts For Your Next Test

1. The product property of roots states that $ extbackslash sqrt[n] extbraceleft ab extbraceright = extbackslash sqrt[n] extbraceleft a extbraceright extbackslash cdot extbackslash sqrt[n] extbraceleft b extbraceright$, where $a$ and $b$ are positive real numbers and $n$ is a positive integer.
2. This property allows you to simplify expressions involving roots of products by breaking them down into the product of individual roots.
3. The product property of roots is useful when working with expressions that contain multiplication of radicands, as it can help reduce the number of radical signs.
4. The product property of roots is also known as the power rule for roots, as it is similar to the power rule for exponents: $a^m extbackslash cdot a^n = a^{m+n}$.
5. Understanding the product property of roots is essential when simplifying and evaluating expressions involving higher roots, such as cube roots or fourth roots.

Review Questions

• Explain how the product property of roots can be used to simplify radical expressions.
  • The product property of roots states that the nth root of a product is equal to the product of the nth roots of the individual factors. This means that if you have an expression like $ extbackslash sqrt extbraceleft 16 extbackslash cdot 9 extbraceright$, you can simplify it by breaking it down into $ extbackslash sqrt extbraceleft 16 extbraceright extbackslash cdot extbackslash sqrt extbraceleft 9 extbraceright$, which is equal to 4 $ extbackslash cdot$ 3 = 12. This property allows you to simplify complex radical expressions by separating the radicands and evaluating the individual roots.
• Describe how the product property of roots is similar to the power rule for exponents.
  • The product property of roots, $ extbackslash sqrt[n] extbraceleft ab extbraceright = extbackslash sqrt[n] extbraceleft a extbraceright extbackslash cdot extbackslash sqrt[n] extbraceleft b extbraceright$, is similar to the power rule for exponents, $a^m extbackslash cdot a^n = a^{m+n}$. Both properties allow you to break down expressions involving products by separating the individual factors. Just as the power rule allows you to combine exponents, the product property of roots allows you to combine roots, making it a useful tool for simplifying and evaluating radical expressions.
• Explain how the product property of roots is essential for working with higher roots, such as cube roots or fourth roots.
  • Understanding the product property of roots is crucial when dealing with higher roots, such as cube roots or fourth roots, because it allows you to simplify and evaluate expressions that contain multiplication of radicands. For example, if you have an expression like $ extbackslash sqrt[4] extbraceleft 16 extbackslash cdot 81 extbraceright$, you can use the product property to rewrite it as $ extbackslash sqrt[4] extbraceleft 16 extbraceright extbackslash cdot extbackslash sqrt[4] extbraceleft 81 extbraceright$, which is equal to 2 $ extbackslash cdot$ 3 = 6. This property is a fundamental tool for working with higher roots and is essential for mastering the concepts in the 9.7 Higher Roots topic.

Related terms

Root: A root is a value that, when raised to a given power, produces a specific number. For example, the square root of 9 is 3, because 3^2 = 9.

Exponent: An exponent is a number that indicates how many times the base number is multiplied by itself. For example, in the expression $x^3$, the exponent is 3, meaning $x$ is multiplied by itself 3 times.

Radical Expression: A radical expression is an expression that contains a root, such as $ extbackslash sqrt extbraceleft 16 extbraceright$ or $ extbackslash sqrt[3] extbraceleft 27 extbraceright$.