In Exercises 45–66, divide and, if possible, simplify. ___ √200 √10

Verified step by step guidance

1. <insert step 1: Start by expressing both the numerator and the denominator in terms of their prime factors.>
2. <insert step 2: The square root of 200 can be expressed as \( \sqrt{200} = \sqrt{2^3 \times 5^2} \).>
3. <insert step 3: The square root of 10 can be expressed as \( \sqrt{10} = \sqrt{2 \times 5} \).>
4. <insert step 4: Divide the expressions: \( \frac{\sqrt{200}}{\sqrt{10}} = \sqrt{\frac{2^3 \times 5^2}{2 \times 5}} \).>
5. <insert step 5: Simplify the expression by canceling out common factors in the numerator and the denominator.>

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, we are dealing with square roots, which represent the value that, when multiplied by itself, gives the original number. Understanding how to manipulate these expressions is crucial for simplifying and dividing them effectively.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form. This often includes factoring out perfect squares from under the radical sign. For example, √200 can be simplified by recognizing that 200 = 100 × 2, allowing us to express √200 as 10√2.

Division of Radicals

Dividing radical expressions requires applying the properties of radicals, particularly the quotient rule, which states that √a / √b = √(a/b). This means we can simplify the division of two square roots by combining them under a single radical, provided that the denominator is not zero.