Conjugate

Definition

The conjugate of a radical expression is a new expression created by changing the sign of the radicand, or the quantity under the radical symbol. Conjugates are particularly important in simplifying and dividing radical expressions.

5 Must Know Facts For Your Next Test

1. The conjugate of $\sqrt{a}$ is $-\sqrt{a}$.
2. Multiplying a radical expression by its conjugate will result in a non-radical expression.
3. Conjugates are used to rationalize the denominator of a fraction with a radical expression in the denominator.
4. When dividing radical expressions, the divisor is multiplied by its conjugate to eliminate the radical in the denominator.
5. Conjugates are essential for simplifying expressions with roots and dividing radical expressions.

Review Questions

• Explain how to find the conjugate of a given radical expression.
  • To find the conjugate of a radical expression, $\sqrt{a}$, you change the sign of the radicand, or the quantity under the radical symbol. So the conjugate of $\sqrt{a}$ would be $-\sqrt{a}$. This is an important step in simplifying expressions with roots and dividing radical expressions, as multiplying by the conjugate can eliminate the radical in the denominator.
• Describe the purpose of using conjugates when dividing radical expressions.
  • When dividing one radical expression by another, the divisor is multiplied by its conjugate to eliminate the radical in the denominator. This allows the division to be performed without a radical in the denominator, which simplifies the expression. For example, to divide $\sqrt{12}$ by $\sqrt{3}$, we would multiply the divisor $\sqrt{3}$ by its conjugate $-\sqrt{3}$, resulting in $\frac{\sqrt{12}}{-\sqrt{3}} = -2$.
• Analyze how conjugates can be used to simplify expressions with roots.
  • Multiplying a radical expression by its conjugate will result in a non-radical expression. This is because the product of a radical expression and its conjugate will cancel out the radical symbol. For instance, $\sqrt{a} \cdot (-\sqrt{a}) = -a$. This property of conjugates is essential for simplifying expressions with roots, as it allows you to eliminate the radical symbol and perform further algebraic operations to arrive at a simpler form.

Related terms

Radicand: The quantity under the radical symbol in a radical expression.

Simplify: To reduce an expression to its simplest form by performing operations and applying properties.

Division of Radicals: The process of dividing one radical expression by another.