Textbook Question
In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers.___√x⁶y⁷
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0. Review of Algebra
Radical Expressions
3:50 minutesProblem 43
Textbook Question
In Exercises 33–44, add or subtract terms whenever possible. 3√8−√32+3√72−√75
Verified step by step guidance1
First, simplify each square root term individually. Start with \( \sqrt{8} \).
Recognize that \( 8 = 4 \times 2 \), so \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \).
Next, simplify \( \sqrt{32} \). Notice that \( 32 = 16 \times 2 \), so \( \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \).
Now, simplify \( \sqrt{72} \). Since \( 72 = 36 \times 2 \), we have \( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \).
Finally, simplify \( \sqrt{75} \). Recognize that \( 75 = 25 \times 3 \), so \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radical Expressions
Radical expressions involve roots, such as square roots or cube roots. Understanding how to simplify these expressions is crucial, as it allows for the combination of like terms. For example, √a and √b can be combined if they share the same radicand, or if they can be simplified to a common form.
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Simplifying Radicals
Simplifying radicals involves breaking down the radicand into its prime factors and extracting perfect squares or cubes. For instance, √32 can be simplified to 4√2, as 32 = 16 × 2. This process is essential for adding or subtracting radical terms effectively.
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Combining Like Terms
Combining like terms is a fundamental algebraic skill that involves adding or subtracting terms that have the same variable or radical part. In the context of radicals, this means only combining terms that are identical in their simplified form, such as 3√2 and 5√2, which can be combined to yield 8√2.
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