Adding and Subtracting Radicals
Adding and Subtracting Radicals explains that like radicals — those with the same radicand — can be combined by adding or subtracting their coefficients, just like combining like terms. Covered in Yoshiwara Elementary Algebra Chapter 9: More About Exponents and Roots, Grade 6 students learn that unlike radicals (different radicands) cannot be combined directly, though sometimes simplifying one or both radicals first reveals hidden like radicals. This skill is essential for simplifying radical expressions.
Key Concepts
Property Square roots with identical radicands are called like radicals . We can add or subtract like radicals in the same way that we add or subtract like terms, namely by adding or subtracting their coefficients. For example, $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$.
Examples Combining like radicals: $9\sqrt{5} 4\sqrt{5} = (9 4)\sqrt{5} = 5\sqrt{5}$. Unlike radicals cannot be combined: $8\sqrt{3} + 2\sqrt{7}$ cannot be simplified into a single term. Sometimes you must simplify first: $\sqrt{18} + \sqrt{50} = \sqrt{9 \cdot 2} + \sqrt{25 \cdot 2} = 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$.
Explanation You can only add or subtract radicals if the number inside the square root (the radicand) is exactly the same. Think of $\sqrt{3}$ as a variable like $x$. You can combine $5x + 2x$ but not $5x + 2y$.
Common Questions
How do you add radicals?
Only like radicals (same radicand and same index) can be added. Add the coefficients and keep the radical part: 3√5 + 2√5 = 5√5.
Can you add √3 and √5?
No, they are unlike radicals with different radicands. They cannot be combined and must be left as √3 + √5.
How do you simplify before adding radicals?
Simplify each radical first. For example, √12 = 2√3 and √27 = 3√3, so √12 + √27 = 2√3 + 3√3 = 5√3.
Where is adding and subtracting radicals in Yoshiwara Elementary Algebra?
This is covered in Chapter 9: More About Exponents and Roots of Yoshiwara Elementary Algebra.
What are like radicals?
Like radicals have the same radicand and the same index (root type). For example, 4√7 and -2√7 are like radicals, both having radicand 7.