Product Property of Square Roots

Definition

The product property of square roots states that the square root of a product is equal to the product of the square roots of the individual factors. This property allows for simplifying and multiplying square roots efficiently.

5 Must Know Facts For Your Next Test

1. The product property of square roots states that $\sqrt{ab} = \sqrt{a}\sqrt{b}$.
2. This property allows for the multiplication of square roots by multiplying the values inside the square root symbols.
3. Simplifying square roots often involves identifying perfect square factors that can be extracted from the radicand.
4. The product property is particularly useful when multiplying expressions containing square roots.
5. Applying the product property can help reduce the complexity of radical expressions and make them easier to evaluate.

Review Questions

• Explain how the product property of square roots can be used to simplify a radical expression.
  • The product property of square roots states that $\sqrt{ab} = \sqrt{a}\sqrt{b}$. This means that if a radical expression contains a product of two or more factors, the square root can be rewritten as the product of the square roots of those factors. By identifying perfect square factors within the radicand, you can extract them and simplify the expression. For example, $\sqrt{72} = \sqrt{4 \times 18} = \sqrt{4}\sqrt{18} = 2\sqrt{18}$.
• Describe how the product property of square roots can be used to multiply square root expressions.
  • When multiplying square root expressions, the product property can be applied to simplify the calculation. The property states that $\sqrt{ab} = \sqrt{a}\sqrt{b}$. This means that to multiply two square root expressions, you can multiply the values inside the square root symbols and then take the square root of the result. For example, to multiply $\sqrt{3}$ and $\sqrt{5}$, you would calculate $\sqrt{3}\sqrt{5} = \sqrt{3 \times 5} = \sqrt{15}$.
• Evaluate the expression $\sqrt{8}\sqrt{18}$ using the product property of square roots.
  • To evaluate the expression $\sqrt{8}\sqrt{18}$ using the product property of square roots, we can rewrite it as $\sqrt{8 \times 18}$. The product property states that $\sqrt{ab} = \sqrt{a}\sqrt{b}$, so we can simplify this further to $\sqrt{8}\sqrt{18} = \sqrt{8 \times 18} = \sqrt{144} = 12$. By applying the product property, we can easily evaluate the expression and arrive at the final answer of 12.

Related terms

Square Root: The square root of a number is the value that, when multiplied by itself, gives the original number.

Radical Expression: A radical expression contains one or more square roots.

Simplification of Square Roots: The process of rewriting a square root expression in its simplest form.