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Chapter 8: Problem 1

In each case, simplify the radical expressions by placing them under the same radical sign. \((\sqrt{3})(\sqrt{3})\)

Short Answer

Expert verified
3

Step by step solution

01

Write the expression

The given expression is \((\sqrt{3})(\sqrt{3})\)
02

Use the property of radicals

According to the property of radicals, \((\sqrt{a})(\sqrt{b}) = \sqrt{ab}\). Applying this property: \((\sqrt{3})(\sqrt{3}) = \sqrt{3 \cdot 3}\)
03

Multiply the radicands

Multiply the numbers under the radical sign: \((\sqrt{3 \cdot 3}) = \sqrt{9}\)
04

Simplify the radical

Find the square root of 9: \((\sqrt{9}) = 3\)

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Key Concepts

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

radicals
Radicals are expressions that include a root symbol. The most common type is the square root, represented as \(\sqrt{a}\). Radicals can also represent cube roots (\sqrt[3]{a}) and higher roots.
Radicals are used to express numbers that can't be simplified into a whole number. For example, \(\sqrt{2}\) is a radical because it represents a number that, when multiplied by itself, equals 2.
In the context of the given exercise, we deal with square roots, which are a basic type of radical.
properties of radicals
To simplify radical expressions, it is important to understand some key properties of radicals:
  • Multiplication Property: \(\sqrt{a} \sqrt{b} = \sqrt{ab}\)
  • Division Property: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)

These properties help us manipulate and simplify expressions involving radicals. For instance, in the given problem, we use the multiplication property to combine \(\sqrt{3}\) and \(\sqrt{3}\) under a single radical sign.
multiplication of radicals
When multiplying radicals, utilize the multiplication property of radicals. In our exercise, we start with \(\sqrt{3} \cdot \sqrt{3}\).
We can apply the property: \(\begin{aligned} \sqrt{a}\cdot\sqrt{b}&=\sqrt{ab}&\text{(Multiplication Property)}bsp;&end{aligned}\)) This turns \(\sqrt{3} \cdot \sqrt{3}\) into \(\sqrt{3\cdot3}\), which simplifies to \(\sqrt{9}\).
simplifying square roots
Once we have a single radical, we often need to simplify it. Simplifying a square root involves finding the number that, when squared, equals the radicand.
For example, \(\sqrt{9}\) asks for the number that, when multiplied by itself, gives 9. Since \(3\cdot3 = 9\), it simplifies to 3.
Thus, the initial expression \(\sqrt{3} \cdot \sqrt{3}\) ultimately simplifies to 3 through these property applications and operations.

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Most popular questions from this chapter

Change the exponential expressions into radical expressions. \(10^{\frac{1}{2}}\)

Simplify the following radical expressions by factoring. \(\sqrt{20}\)

Solve the radical expressions by finding the positive roots. \(\sqrt[3]{64}\)

In each of the following, add or subtract the radical expressions. \(5 \sqrt[4]{x}+2 \sqrt[4]{x}\)

Solve the following radicals. Be careful. Some of them are undefined. \(\sqrt[5]{-32}\)
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