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Adding and Subtracting Square Roots that Need Simplification
When adding or subtracting square roots, the terms may not initially appear to be like radicals because their radicands differ. However, after simplifying each square root by removing the largest perfect-square factor, the simplified forms may reveal like radicals that can then be combined.
The general strategy has two stages:
- Simplify each radical by factoring out the largest perfect square from the radicand and applying the Product Property.
- Combine like radicals by adding or subtracting their coefficients, just as one combines like terms in algebra.
When a radical already has a numerical coefficient — for example, — simplifying the radical produces a product of three factors (such as ). The Associative Property of Multiplication allows these numerical factors to be multiplied together first, yielding a single coefficient in front of the radical (). This mirrors the algebraic simplification , where the two numerical factors are combined while the variable part remains unchanged.
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Adding and Subtracting Like Radicals
Adding and Subtracting Square Roots that Need Simplification
Simplifying
Which pair of square roots are 'like' and can be combined by adding?
Match each square root expression with its simplified result.
Simplify: _____
Arrange the steps in order to simplify .
In , what coefficient does have when no number is written in front of it?
Simplify . Explain what happens to the coefficients and what happens to the radicand.
What makes two square roots "like," and how do you add or subtract them?
Which rule applies when adding or subtracting square roots?
True or False:
Simplify . What happens to the coefficients, and what happens to the radical part?
Learn After
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Simplifying
Simplifying
Simplifying
Simplifying
Match each square root expression with its simplified form.
To simplify , factor out the largest _____ from the radicand.
How do you add like radicals, such as ?
True or False: Before adding and , you should simplify each radical to see if they become like radicals.
Put the steps in order to simplify .
After simplifying, two square roots are called "like radicals" when they share which feature?
Simplifying gives , and then 6 and 2 are multiplied to get . Which property justifies multiplying 6 and 2 together?