Remember that for the rest of this chapter, it will be assumed that all variables represent non-negative real numbers.
So, for example
 = a, a  0.
- Product Rule For Radicals
- For non-negative real numbers a and b,
- which holds since
- Simplifying Radicals Whose Radicands Are Natural Numbers
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- Write the radicand as the product of two numbers, one of which is the largest perfect power number for the given index.
- Use the product rule to write the expression as a product of roots.
- Find the roots of any perfect power numbers. This means, simplify.
- You are done when the radicand has no factor that is a perfect power.
- You obtain the numbers that are perfect powers by squaring the natural numbers.
- Simplifying Radicals Whose Radicands Are Variables
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- Write the radicand as the product of two numbers, one of which is the largest perfect power of the variable for the given index.
- Use the product rule to write the expression as a product of radicals.
Place all perfect powers under the same radical.
- Find the roots of any perfect powers. This means, simplify.
- You are done when the radicand has no more perfect powers of any variable for the given index.
- Simplifying Radicals
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- If the radicand contatins a numerical factor, write it as a product of two numbers, one of which is the largest perfect power number for the given index.
- Write the radicand as the product of two numbers, one of which is the largest perfect power of the variable for the given index.
- Use the product rule to write the expression as a product of radicals.
Place all perfect powers (numbers and variables) under the same radical.
- Find the roots of any perfect powers. This means, simplify.
- You are done when the radicand has no factor of the number that is a perfect power and no more perfect powers of any variable for the given index.
- Multiplying Radicals
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- Multiply radicals using the product rule.
- After multiplying, use the steps to simplify radicals stated above.
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