How to change a mixed radical to an entire radical and an entire radical to a mixed radical

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Welcome to our lesson on converting between mixed and entire radicals. A mixed radical has a coefficient outside the radical sign, like 3 square root of 5. An entire radical has everything under the radical sign, like square root of 45. These two forms are equivalent and we can convert between them using specific methods. Now let's learn how to convert a mixed radical to an entire radical. The formula is: a times the nth root of b equals the nth root of a to the n times b. Let's work through an example with 3 square root of 5. First, identify the coefficient a equals 3, the index n equals 2, and the radicand b equals 5. Next, calculate a to the n, which is 3 squared equals 9. Then multiply a to the n times b, which gives us 9 times 5 equals 45. Finally, place this result under the radical to get square root of 45. Now let's learn the reverse process: converting an entire radical to a mixed radical. The key is finding the largest perfect nth power factor. Let's work with square root of 48. First, we need to find the largest perfect square factor of 48. The perfect squares are 1, 4, 9, 16, 25, and so on. The largest perfect square that divides 48 is 16, since 48 equals 16 times 3. We rewrite square root of 48 as square root of 16 times 3. Using the property of radicals, we separate this into square root of 16 times square root of 3. Since square root of 16 equals 4, our final answer is 4 square root of 3. Let's practice with cube roots. First, let's convert 2 cube root of 7 to an entire radical. We identify a equals 2, n equals 3, and b equals 7. Calculate a cubed: 2 cubed equals 8. Multiply 8 times 7 to get 56. So 2 cube root of 7 equals cube root of 56. Now for the reverse: convert cube root of 108 to mixed form. We need the largest perfect cube factor of 108. Since 108 equals 27 times 4, and 27 equals 3 cubed, we get 3 cube root of 4. These examples show how the same principles apply to any index. Let's summarize the key concepts. To convert from mixed to entire radical, use the formula: a times the nth root of b equals the nth root of a to the n times b. To convert from entire to mixed radical, find the largest perfect nth power factor and extract it. For example, 5 square root of 2 becomes square root of 50, while square root of 72 becomes 6 square root of 2. Remember to practice identifying perfect powers: squares like 1, 4, 9, 16, and cubes like 1, 8, 27, 64. These conversions are fundamental skills in algebra and will help you simplify radical expressions efficiently.