Binomial Radical Expression

Definition

A binomial radical expression is an algebraic expression that contains two terms, one of which is a radical (square root) term. These expressions involve the addition, subtraction, or multiplication of a rational number or variable with a square root of a rational number or variable.

5 Must Know Facts For Your Next Test

1. Binomial radical expressions can be simplified by combining like terms and applying the properties of radicals.
2. Adding or subtracting binomial radical expressions involves combining the rational number or variable terms and the radical terms separately.
3. Multiplying binomial radical expressions involves distributing the multiplication and simplifying the resulting radical terms.
4. Dividing binomial radical expressions involves rationalizing the denominator, which may require the use of conjugates.
5. Binomial radical expressions can be used to model and solve a variety of real-world problems, such as those involving areas, volumes, and other geometric quantities.

Review Questions

• Explain the process of simplifying a binomial radical expression by combining like terms and applying the properties of radicals.
  • To simplify a binomial radical expression, you first need to identify the like terms, which are the terms that have the same radicand (the number under the radical sign). You can then combine the like terms by adding or subtracting the coefficients of the radical terms. Additionally, you can apply the properties of radicals, such as $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ and $\sqrt{a} / \sqrt{b} = \sqrt{a/b}$, to further simplify the expression.
• Describe the steps involved in adding or subtracting binomial radical expressions.
  • To add or subtract binomial radical expressions, you need to first identify the like terms, which are the terms that have the same radicand. You can then add or subtract the coefficients of the like terms separately. For example, to add $3\sqrt{2} + 5\sqrt{2}$, you would combine the like terms by adding the coefficients: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$. The same process applies to subtracting binomial radical expressions, where you would subtract the coefficients of the like terms.
• Analyze the steps required to multiply binomial radical expressions, including the use of the distributive property and simplification of the resulting radical terms.
  • To multiply binomial radical expressions, you need to use the distributive property to multiply each term of the first expression with each term of the second expression. This will result in a quadrinomial expression, which can then be simplified by combining like terms and applying the properties of radicals. For example, to multiply $(2\sqrt{3}) + (4\sqrt{5})$ by $(3\sqrt{3}) - (\sqrt{5})$, you would first distribute the multiplication: $(2\sqrt{3} + 4\sqrt{5})(3\sqrt{3} - \sqrt{5}) = 6\sqrt{9} - 2\sqrt{15} + 12\sqrt{15} - 4\sqrt{25}$. You can then simplify this expression by combining like terms and applying the properties of radicals: $6\sqrt{9} - 2\sqrt{15} + 12\sqrt{15} - 4\sqrt{25} = 18\sqrt{3} + 10\sqrt{15} - 4\sqrt{25} = 18\sqrt{3} + 10\sqrt{15} - 20$.

Related terms

Radical Expression:

A radical expression is an algebraic expression that contains a square root symbol (√) or a higher-order root symbol.

Rational Number: A rational number is a number that can be expressed as a fraction, where the numerator and denominator are integers, and the denominator is not zero.

Variable:

A variable is a symbol, usually a letter, that represents an unknown or changeable quantity in an algebraic expression.