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Chapter 9: Problem 3

Write the expression as a product of two radicals and simplify. $$\sqrt{64 \cdot 11}$$

Short Answer

Expert verified
The simplified expression is \(8 \cdot \sqrt{11}\)

Step by step solution

01

Split the square root

The given expression is \(\sqrt{64 \cdot 11}\). According to the properties of square roots, the square root of a product can be written as the product of the square roots. Hence, it can be written as \(\sqrt{64} \cdot \sqrt{11}\).
02

Simplify the square root

The square root of 64 is 8, since \(8^2 = 64\), and the square root of a perfect square is the original number. Therefore, the expression becomes \(8 \cdot \sqrt{11}\).

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Key Concepts

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

Understanding Square Roots
A square root of a number is a value that, when multiplied by itself, results in the original number. For example, the square root of 64 is 8 because when you multiply 8 by itself, you get 64. Square roots are denoted by the radical symbol, \(\sqrt{}\).
Understanding how square roots work is crucial in algebra, especially when simplifying expressions. Here are some key points to remember:
  • Square roots can apply to both perfect squares (like 64) and non-perfect squares (such as 11).
  • The square root of perfect squares is whole numbers, which can simplify calculations.
  • If the number is not a perfect square, its square root remains in radical form (e.g., \(\sqrt{11}\)).
When working with square roots, it's essential to identify if the number has a whole number root, which makes it easier to simplify the expression further.
Product of Radicals
The concept of the product of radicals stems from the property that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). This property allows us to break down complex expressions into simpler parts. This is particularly useful when at least one of the numbers involved is a perfect square.
Consider the example \(\sqrt{64 \cdot 11}\). Using the product of radicals property, we can split this into two separate radicals: \(\sqrt{64} \cdot \sqrt{11}\).
Some advantages of this property include:
  • It simplifies calculations by isolating perfect squares as integers.
  • It helps in breaking down and simplifying complex expressions much easier.
Always be sure to use this property with numbers inside the radical to ensure your solutions are as simple as possible.
Simplifying Expressions
Simplifying expressions involves rewriting them in a more concise and manageable form. When dealing with expressions that include radicals, like square roots, simplifying often means finding any perfect squares and reducing the expression to its simplest terms.
Let's consider the expression \(\sqrt{64} \cdot \sqrt{11}\). Because 64 is a perfect square, we know that \(\sqrt{64}\) simplifies straight to 8. This reduces the expression to \(8 \cdot \sqrt{11}\).
In terms of simplifying expressions, keep in mind:
  • Look for and separate any perfect squares within the radical, since they simplify directly to integers.
  • The goal is to make the expression as simple as possible without changing its value, sometimes leaving non-perfect squares in radical form.
By consistently applying these key practices, simplifying expressions with radicals will become second nature.

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