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Chapter 15: Problem 12

Simplify. $$\sqrt{11} \cdot \sqrt{11}$$

Short Answer

Expert verified
The simplified result is 11.

Step by step solution

01

Identify terms under the square root

Identify the terms under the square root. In this case, the term is 11 in both square roots.
02

Multiply the terms

Multiply the terms under the square root sign. Since they are the same (i.e. 11), this can be calculated as \( \sqrt{11^2} \).
03

Calculate the square root

Calculate the square root of the resulting number, which is \( \sqrt{11^2}= \sqrt{121} \).

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Key Concepts

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

Square Root Multiplication
Multiplying square roots is a simple and useful operation. When dealing with square roots, you multiply the numbers under the radical signs as if they were regular numbers. The formula for multiplying two square roots \( \sqrt{a} \cdot \sqrt{b} \) results in \( \sqrt{a \cdot b} \). This rule makes it straightforward to combine square roots.
For example, in the exercise \( \sqrt{11} \cdot \sqrt{11} \), both terms under the square roots are 11. By multiplying under the radical, you get \( \sqrt{11 \times 11} = \sqrt{11^2} \). Simply focus on combining the terms inside the square root before simplifying.
Properties of Square Roots
Understanding the properties of square roots allows you to perform calculations effortlessly. One key property is that the square root of a number squared is simply the original number. Mathematically, it is represented as \( \sqrt{a^2} = a \). This property is invaluable when simplifying square roots that result from multiplication.
When you solve \( \sqrt{11} \cdot \sqrt{11} \), you apply this property. After obtaining \( \sqrt{11^2} \), the square root naturally undoes the squaring of 11, yielding \( 11 \).
  • Know that \( \sqrt{a^2} = a \) helps eliminate the square roots in most cases.
  • This property turns the task of simplifying into recognizing patterns.
Radical Expressions
Radical expressions involve variables or numbers under a root symbol. Simplifying these expressions can make them easier to work with, especially when they include operations like addition, subtraction, multiplication, or division.
A crucial aspect of working with radicals is knowing when and how to simplify. After multiplying radicands (the numbers inside the square root), check if the product is a perfect square. If it is, simplify further as in the example \( \sqrt{11} \cdot \sqrt{11} = \sqrt{121} \), where 121 is a perfect square. The result simplifies neatly to \( 11 \).
  • Simplify by multiplying and then checking for perfect squares.
  • Reduces clutter and can transform complex-looking problems into simple solutions.
Radical expressions are common in algebra, and understanding them enhances problem-solving skills.

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