Like Radicals

Definition

Like radicals are square roots, cube roots, or other radical expressions that have the same index and are raised to the same power. They can be combined through addition, subtraction, or multiplication to simplify radical expressions.

5 Must Know Facts For Your Next Test

1. Like radicals can be added or subtracted by combining the coefficients and keeping the same radical expression.
2. Multiplying like radicals involves multiplying the coefficients and adding the exponents of the radical expression.
3. Dividing like radicals involves dividing the coefficients and subtracting the exponents of the radical expression.
4. Raising a like radical to a power involves raising the coefficient to that power and multiplying the exponent by that power.
5. Simplifying a radical expression often involves identifying and combining like radicals.

Review Questions

• Explain how to add or subtract like radicals and provide an example.
  • To add or subtract like radicals, you combine the coefficients while keeping the same radical expression. For example, $\sqrt{5} + 3\sqrt{5} = 4\sqrt{5}$, and $7\sqrt{3} - 2\sqrt{3} = 5\sqrt{3}$. The key is that the radicals must have the same index and be raised to the same power in order to be considered like radicals.
• Describe the process of multiplying like radicals and explain how it differs from adding or subtracting them.
  • When multiplying like radicals, you multiply the coefficients and add the exponents of the radical expressions. For instance, $\sqrt{2} \cdot 3\sqrt{2} = 3\sqrt{2^2} = 3\sqrt{4} = 6\sqrt{2}$. This is different from adding or subtracting like radicals, where you only combine the coefficients. Multiplying like radicals involves a more complex operation that takes into account the exponents of the radical expressions.
• Analyze how simplifying radical expressions often requires identifying and combining like radicals. Provide an example to illustrate this concept.
  • Simplifying radical expressions frequently involves recognizing and combining like radicals. For example, to simplify the expression $\sqrt{8} + 2\sqrt{2} - \sqrt{18} + 3\sqrt{2}$, we first identify the like radicals: $\sqrt{8}$ and $\sqrt{2}$. We can then combine them by adding the coefficients: $\sqrt{8} + 2\sqrt{2} + 3\sqrt{2} = \sqrt{8} + 5\sqrt{2}$. Recognizing and combining like radicals is a crucial step in simplifying more complex radical expressions.

Related terms

Radical Expression: A mathematical expression containing one or more radicals, such as $\sqrt{x}$ or $\sqrt[3]{y}$.

Radical Index: The number that indicates the root being taken, such as 2 for a square root or 3 for a cube root.

Rational Exponent: An exponent that can be expressed as a fraction, such as $x^{2/3}$, which is equivalent to $\sqrt[3]{x^2}$.