Use the product rule to simplify the expressions in Exercises 41 - 44. In exercises 43 - 44, assume that variables represent nonnegative real numbers. √300

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Recognize that the expression $ \sqrt{300} $ can be rewritten using exponents as $ 300^{\frac{1}{2}} $.

Factor 300 into its prime factors or into a product of a perfect square and another number. For example, $ 300 = 100 \times 3 $.

Use the property of radicals that $ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $ to rewrite $ \sqrt{300} $ as $ \sqrt{100} \times \sqrt{3} $.

Simplify $ \sqrt{100} $ since 100 is a perfect square, which equals 10, so the expression becomes $ 10 \times \sqrt{3} $.

Express the simplified form as $ 10\sqrt{3} $, which is the simplified radical form of $ \sqrt{300} $.

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Product Rule for Radicals

The product rule for radicals states that the square root of a product equals the product of the square roots: √(a * b) = √a * √b. This rule helps simplify expressions by breaking down numbers into factors whose roots are easier to compute.

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Expanding Radicals

Simplifying Square Roots

Simplifying square roots involves factoring the radicand into perfect squares and other factors, then taking the square root of the perfect squares outside the radical. For example, √300 can be simplified by factoring 300 into 100 * 3, giving 10√3.

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Imaginary Roots with the Square Root Property

Nonnegative Real Numbers Assumption

Assuming variables represent nonnegative real numbers ensures that square roots are defined as principal (nonnegative) roots. This assumption allows the use of the product rule without considering complex numbers or negative roots.

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