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Chapter 0: Problem 121

Explain how to simplify \(\sqrt{10} \cdot \sqrt{5}\).

Short Answer

Expert verified
The simplified form of \(\sqrt{10} \cdot \sqrt{5}\) is \(5\sqrt{2}\).

Step by step solution

01

Recognize the square root property

For non-negative real numbers \(a\) and \(b\), \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)}. This means that \(\sqrt{10} \cdot \sqrt{5}\) can be rewritten as \(\sqrt{10 \cdot 5}\).
02

Calculate the product inside the square root.

Calculate \(10 \cdot 5\) to get 50. So, \(\sqrt{10} \cdot \sqrt{5} = \sqrt{50}\) essentially.
03

Simplify the Square root

The number 50 can be expressed as \(25 \times 2\) or \((5^2) \times 2 \). We know that the square root of \(5^2\) is 5. So we can simplify \(\sqrt{50}\) to \(5\sqrt{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Property
When you're working with square roots, understanding the square root property is essential. This property allows you to simplify products of square roots by combining them into a single square root.
For any non-negative real numbers \(a\) and \(b\), the square root property states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
Let's apply this property. Consider \(\sqrt{10} \cdot \sqrt{5}\).
  • First, recognize that according to the square root property, you can combine these as one square root: \(\sqrt{10 \cdot 5}\).
  • This transformation helps in reducing multiple radical expressions into a single one, thereby simplifying the expression significantly.

Keep in mind that the square root property only applies to non-negative numbers. Ensuring this condition helps to avoid undefined operations in real numbers.
Product of Radicals
When simplifying expressions involving radicals, calculating the product inside the square root is our next logical step. This involves straightforward multiplication of the numbers under the square roots.
Continuing from our example, after applying the square root property, we have \(\sqrt{10 \cdot 5}\).
  • This becomes \(\sqrt{50}\) when you multiply 10 and 5.
  • The task now is to see if we can break \(\sqrt{50}\) down further using its prime factors.

Understanding and applying the concept of products of radicals allows for further simplification by reducing their complexity. It transforms separate radical terms into a single, more manageable term.
Simplification of Square Roots
The final step in simplifying radical expressions is breaking down the square root into its simplest form. To do this effectively, we use prime factorization and identify perfect squares within the product.
Taking \(\sqrt{50}\), we break it down as \(25 \times 2\). Since 25 is a perfect square, our focus is on simplifying this part.
  • Recognize that 25 is \(5^2\). The square root of \(5^2\) is 5.
  • Hence, \(\sqrt{50} = \sqrt{(5^2 \times 2)}\) can be reduced to \(5\sqrt{2}\).

This kind of simplification simplifies the expression further, making it more elegant and useful for various mathematical operations. Remember to always look for perfect squares when simplifying roots, as this will guide your simplification effectively. Prefer simplicity in expressions to make further calculations easier.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathbf{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

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If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if \(h,\) the number of outcomes that result in heads, satisfies \(\left|\frac{h-50}{5}\right| \geq 1.645 .\) Describe the number of outcomes that determine an unfair coin that is tossed 100 times.
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