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Chapter 9: Problem 37

Simplify each expression by performing the indicated operation. $$ \sqrt{2}(\sqrt{3}+1) $$

Short Answer

Expert verified
Question: Simplify the expression: \(\sqrt{2}(\sqrt{3} + 1)\). Answer: \(\sqrt{6} + \sqrt{2}\).

Step by step solution

01

Apply the distributive property

Distribute the \(\sqrt{2}\) to both terms inside the parentheses. Remember, the distributive property allows us to multiply \(\sqrt{2}\) by both terms inside the parentheses individually: $$ \sqrt{2}(\sqrt{3} + 1) = \sqrt{2} \cdot \sqrt{3} + \sqrt{2} \cdot 1 $$
02

Multiply the radical numbers

Now, multiply the radical numbers together: $$ \sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} $$ This simplifies to: $$ \sqrt{6} $$
03

Multiply the radical number and the integer

Now, multiply \(\sqrt{2}\) by the integer 1: $$ \sqrt{2} \cdot 1 = \sqrt{2} $$
04

Combine the results

Finally, combine the results from steps 2 and 3: $$ \sqrt{6} + \sqrt{2} $$ The simplified expression is: $$ \sqrt{6} + \sqrt{2} $$

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Key Concepts

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

Distributive Property
The distributive property is a fundamental principle in algebra that allows us to simplify expressions by distributing a multiplier over terms within parentheses. For example, in the expression \(a(b + c)\), the distributive property lets us multiply \(a\) by both \(b\) and \(c\), resulting in \(ab + ac\).

When applied to radical expressions, the same principle holds. In the exercise \(\sqrt{2}(\sqrt{3}+1)\), the multiplier \(\sqrt{2}\) is distributed to both \(\sqrt{3}\) and 1, which simplifies to \(\sqrt{2} \cdot \sqrt{3} + \sqrt{2} \cdot 1\). Understanding and applying the distributive property is essential for simplifying such algebraic expressions.
Multiplying Radicals
Multiplying radicals, numbers under the square root, requires an understanding of a key concept: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). This is possible when both radicals have the same index, commonly index 2 for square roots.

In our exercise, we multiply \(\sqrt{2}\) by \(\sqrt{3}\) to get \(\sqrt{2 \cdot 3}\), which simplifies to \(\sqrt{6}\). Remember, this simplification rule only works if the radicands鈥攖he numbers inside the radical鈥攁re not negative, since the square root of a negative number introduces complex numbers, which are beyond the scope of elementary algebra.
Elementary Algebra
Elementary algebra is the branch of mathematics that deals with variables and constants, operations on them, and the rules that govern these operations. This includes operations with real numbers, radicals, exponents, and the application of the distributive property as shown earlier.

The subject lays the foundation for more advanced topics in mathematics and science. Key skills include simplifying expressions, solving equations, factoring, and manipulating polynomials. The exercise given demonstrates the simplicity and the importance of these basic algebraic skills in action.
Radical Numbers
Radical numbers, or radicals, involve the root of a number, most commonly the square root. The square root of a number \(x\) is a number \(y\) such that \(y^2 = x\). Radicals can be simplified if the number under the radical has a square factor.

For instance, \(\sqrt{18}\) can be simplified to \(3\sqrt{2}\), because 18 is 9 times 2, and the square root of 9 is 3. The exercise we have does not require simplification of the individual radicals \(\sqrt{6}\) and \(\sqrt{2}\), but recognizing when and how to simplify radical numbers can greatly help in solving more complex algebraic problems.

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Most popular questions from this chapter

For the following problems, simplify the expressions. $$ 5 \sqrt{60}-7 \sqrt{15} $$

Simplify each expression by performing the indicated operation. $$ \frac{\sqrt{5}}{3+\sqrt{3}} $$

Simplify each expression by performing the indicated operation. $$ (3 \sqrt{8}-2 \sqrt{2})(4 \sqrt{2}-5 \sqrt{8}) $$

Simplify each expression by performing the indicated operation. $$ \sqrt{8}(\sqrt{3}+\sqrt{2}) $$

For the following problems, simplify each of the square root expressions. $$ \frac{6-\sqrt{3}}{4+\sqrt{2}} $$
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