Explain how the process of combining radicals through addition and subtraction is similar to combining polynomials. What make two radicals like radicals

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Combining radicals through addition and subtraction is similar to combining polynomials because both involve combining like terms. In both cases, we look for terms that have the same variable(s) and the same exponent(s) to simplify the expression.

To understand what makes two radicals "like" radicals, we need to consider the radical expression itself. A radical is an expression that includes a root, such as a square root (√), cube root (∛), or any higher nth root (√ᵢ). In a radical expression, we have a radicand, which is the number or expression inside the radical symbol.

Two radicals are considered like radicals when they have the same index and the same radicand. The index refers to the type of root, such as a square root (index 2), cube root (index 3), and so on. The radicand is the value under the root symbol.

For example, consider the expressions √2 + √3 and 2√2 - 3√3. Both of these expressions consist of square roots and have the same radicands (2 and 3). Hence, the radicals in each expression are considered like radicals.

When combining like radicals through addition or subtraction, we can treat them similarly to combining like terms in polynomials. We perform the same operation (addition/subtraction) on the coefficients (numbers in front of the radical symbol) while keeping the radicals unchanged. For example:

√2 + √3 = (√2) + (√3) = √2 + √3
2√2 - 3√3 = (2√2) + (-3√3) = 2√2 - 3√3

In each case, we add/subtract the coefficients (2 and -3) while keeping the radical expressions (√2 and √3) unchanged.

In summary, combining radicals through addition and subtraction is similar to combining polynomials because both involve combining like terms. Two radicals are considered like radicals when they have the same index and the same radicand.