Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. √7 • √28

Verified step by step guidance

1. Identify the expression to simplify: \(\sqrt{7} \cdot \sqrt{28}\).
2. Use the product rule for radicals, which states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), to combine the radicals into one: \(\sqrt{7 \cdot 28}\).
3. Multiply the numbers inside the radical: \$7 \cdot 28$.
4. Express the product inside the radical as a product of a perfect square and another number to simplify further, for example, find factors of \$7 \cdot 28$ that include a perfect square.
5. Rewrite the radical as the product of the square root of the perfect square and the square root of the remaining factor, then simplify the square root of the perfect square.

Key Concepts

Properties of Radicals

Radicals follow specific rules, such as the product rule which states that the square root of a product equals the product of the square roots: √a • √b = √(a•b). This property allows simplification of expressions involving multiplication under radicals.

Simplifying Radicals

Simplifying radicals involves expressing the number under the root as a product of perfect squares and other factors. For example, √28 can be rewritten as √(4•7), which simplifies to 2√7, making further operations easier.

Assumption of Positive Real Numbers

Assuming variables represent positive real numbers ensures that the principal square root is considered, avoiding ambiguity with negative roots. This assumption allows direct application of radical rules without considering absolute values.