Question

Multiplication of Radicals. Multiply and simplify. $$\sqrt{8} \text { by } \sqrt{160}$$

Short answer

The simplified result of multiplying \(\sqrt{8}\) by \(\sqrt{160}\) is \(8\sqrt{5}\).

Step-by-step solution

Factor into Prime Factors

Factor both radicands into prime factors. For 8, the prime factors are 2 x 2 x 2. For 160, the prime factors are 2 x 2 x 2 x 2 x 2 x 5.

Rewrite Radicals with Prime Factors

Rewrite both radicals using their prime factors: \(\sqrt{8} = \sqrt{2 \times 2 \times 2}\) and \(\sqrt{160} = \sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 5} \).

Multiply Radicals

Use the property \(\sqrt{a}\sqrt{b} = \sqrt{ab}\) to multiply the two radicals: \(\sqrt{8} \times \sqrt{160} = \sqrt{2 \times 2 \times 2} \times \sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 5}\).

Combine and Simplify Inside the Radical

Combine the insides of the radicals and then simplify: \(\sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5} = \sqrt{2^{6} \times 5} = \sqrt{2^{2} \times 2^{2} \times 2^{2} \times 5}\).

Extract Pairs of Prime Factors

Each pair of prime factors outside the radical becomes a single factor, since \(\sqrt{a^{2}}=a\). Therefore, extract the three pairs of 2's as single 2's: \(2 \times 2 \times 2 \times \sqrt{5}\).

Final Simplification

Multiply the extracted factors and leave the remaining prime number inside the radical: \(2 \times 2 \times 2 = 8\) and our expression becomes \(8\sqrt{5}\).

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into a product of its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, the prime factors of 8 are 2 x 2 x 2, since 2 is the only prime number that can evenly divide 8.

When performing prime factorization for the purpose of simplifying radicals, it's essential to find all prime factors. This process helps in identifying pairs of prime factors, which are crucial for simplifying square roots, as replicated pairs can be taken out of the radical sign. Prime factorization is the foundation for simplifying expressions involving radicals, as we will explore further in radical simplification and square roots.

Radical Simplification

Radical simplification involves reducing a radical expression to its simplest form. This often involves using the prime factorization of the number under the radical to group like factors and simplify the overall expression. For example, in our exercise, the number 160 inside the radical is simplified by expressing it as a product of its prime factors, which then reveals pairs of factors that can be brought out of the radical.

This process is a key step in simplifying expressions involving multiplication of radicals. By breaking down each radical to its prime factors, as we did with \(\sqrt{8}\) and \(\sqrt{160}\), we prepare them for multiplication according to the property of radicals that states \(\sqrt{a} \sqrt{b} = \sqrt{ab}\). Simplifying multiplication of radicals, then, becomes a matter of prime factorization followed by the combination and simplification of like factors.

Simplifying Square Roots

Simplifying square roots is a process that follows prime factorization and involves taking pairs of identical prime factors out from under the radical sign. A square root, represented by \(\sqrt{}\), is an operation that asks, 'What number multiplied by itself gives the value inside the square root?'. In simplification, any pair of identical factors (such as \(2 \times 2\)) can be pulled outside the radical as a single factor (just the \(2\) in this case).

For example, in our exercise after combining the radicals, we get \(\sqrt{2^{6} \times 5}\). Notice how \(2^{6}\) has three pairs of twos, so we can bring out three twos, reducing it to \(2 \times 2 \times 2\). The process leaves us with a simpler expression, in this case, \(8\sqrt{5}\), removing the complexity of the larger radical without changing the value of the mathematical expression.