Add or subtract terms whenever possible. $3\sqrt{8} - \sqrt{32} + 3\sqrt{72} - \sqrt{75}$

Verified step by step guidance

1. First, rewrite each radical term by factoring the radicand into a product of a perfect square and another factor. For example, express \(\sqrt{8}\) as \(\sqrt{4 \times 2}\), \(\sqrt{32}\) as \(\sqrt{16 \times 2}\), \(\sqrt{72}\) as \(\sqrt{36 \times 2}\), and \(\sqrt{75}\) as \(\sqrt{25 \times 3}\).
2. Next, simplify each radical by taking the square root of the perfect square factor out of the radical. This means \(\sqrt{4 \times 2} = 2\sqrt{2}\), \(\sqrt{16 \times 2} = 4\sqrt{2}\), \(\sqrt{36 \times 2} = 6\sqrt{2}\), and \(\sqrt{25 \times 3} = 5\sqrt{3}\).
3. Apply the coefficients outside the radicals to the simplified terms. For example, \$3\sqrt{8}\( becomes \)3 \times 2\sqrt{2} = 6\sqrt{2}\(, and \)3\sqrt{72}\( becomes \)3 \times 6\sqrt{2} = 18\sqrt{2}$.
4. Rewrite the entire expression with the simplified terms: \$6\sqrt{2} - 4\sqrt{2} + 18\sqrt{2} - 5\sqrt{3}$.
5. Finally, combine like terms by adding or subtracting the coefficients of the radicals with the same radicand. Combine the \(\sqrt{2}\) terms and leave the \(\sqrt{3}\) term as is, since it has no like terms.

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing the radicand as a product of perfect squares and other factors to rewrite the radical in simplest form. For example, √32 can be simplified to 4√2 because 32 = 16 × 2 and √16 = 4. This process makes it easier to combine like terms.

Like Radicals

Like radicals have the same radicand and index, allowing them to be added or subtracted similarly to like terms in algebra. For instance, 3√2 and 5√2 can be combined as 8√2. Identifying like radicals is essential before performing addition or subtraction.

Combining Like Terms

Combining like terms means adding or subtracting coefficients of terms that have the same variable or radical part. After simplifying radicals, terms with identical radicals can be combined by adding or subtracting their coefficients, simplifying the expression further.