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Chapter 8: Problem 77

Simplify. $$(\sqrt{x}+\sqrt{15})(\sqrt{x}+\sqrt{21})$$

Short Answer

Expert verified
x + \sqrt{21x} + \sqrt{15x} + \sqrt{315}

Step by step solution

01

- Use the distributive property

Apply the distributive property: \( (\sqrt{x} + \sqrt{15})(\sqrt{x} + \sqrt{21}) = \sqrt{x} \cdot \sqrt{x} + \sqrt{x} \cdot \sqrt{21} + \sqrt{15} \cdot \sqrt{x} + \sqrt{15} \cdot \sqrt{21} \)
02

- Simplify each term

Simplify each of the terms separately: \( \sqrt{x} \cdot \sqrt{x} = x \) \( \sqrt{x} \cdot \sqrt{21} = \sqrt{21x} \) \( \sqrt{15} \cdot \sqrt{x} = \sqrt{15x} \) \( \sqrt{15} \cdot \sqrt{21} = \sqrt{315} \)
03

- Combine all terms

Combine the simplified terms together: \( x + \sqrt{21x} + \sqrt{15x} + \sqrt{315} \)

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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 crucial algebraic principle that allows you to multiply a single term by each term within a parenthesis. This is what we do in the first step of the exercise:

When you have an expression like \(a(b + c)\), you distribute \a\ to both \(b\) and \(c\). In the exercise, we distribute \(\sqrt{x}+\sqrt{15}\) into \(\sqrt{x}+\sqrt{21}\). This becomes:
  • \(\sqrt{x} \cdot\sqrt{x}\)
  • \(\sqrt{x}\cdot\sqrt{21}\)
  • \(\sqrt{15}\cdot\sqrt{x}\)
  • \(\sqrt{15}\cdot\sqrt{21}\)

After distributing, you simplify each term.
Radicals
Radicals (square roots) are expressions that involve roots, primarily square roots for beginners. When simplifying expressions with radicals, remember these basic rules:
  • \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\)
  • \(\sqrt{a^2}=a\) if \a\ is non-negative

Referring to the exercise, the product of two square roots, \(\sqrt{x}\cdot\sqrt{21}\), simplifies to \(\sqrt{21x}\). Similarly, \(\sqrt{x}\cdot\sqrt{x}=x\), because \(\sqrt{x^2}=x\) for non-negative \x\.
Combining Like Terms
Combining like terms involves simplifying expressions by merging the terms that have the same variables and exponents. In the context of our exercise:

Though \(x\), \(\sqrt{21x}\), \(\sqrt{15x}\), and \(\sqrt{315}\) are different types of terms, our step here is ensuring all simplified parts are neatly put together. Each term has a different radical or variable part, hence they remain as is without further combining. We see this in the final simplification as: \( x + \sqrt{21x} + \sqrt{15x}+\sqrt{315}\)
Multiplication of Square Roots
When multiplying square roots together, the product of the roots is the root of the products: \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\)
This is key in step 2 of the exercise. For example:
  • \(\sqrt{15}\cdot\sqrt{x}=\sqrt{15x}\)
  • \(\sqrt{15}\cdot\sqrt{21}=\sqrt{315}\)

This principle helps us simplify complex radical expressions efficiently during multiplication.

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