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Chapter 15: Problem 27

Simplify. $$\sqrt{x}(\sqrt{x}-\sqrt{y})$$

Short Answer

Expert verified
The simplified form of \(\sqrt{x}(\sqrt{x}-\sqrt{y})\) is \(x - \sqrt{x*y}\)

Step by step solution

01

Apply the Distributive Property

Distribute \(\sqrt{x}\) to both terms inside the parentheses. That will give: \(\sqrt{x}*\sqrt{x} - \sqrt{x}*\sqrt{y}\)
02

Simplify the First Term

Simplify \(\sqrt{x}*\sqrt{x}\) which is equivalent to square root of \(x^2\), and that gives just x.
03

Simplify the Result

After simplifying, the final result will be: \(x - \sqrt{x*y}\)

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Key Concepts

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

Distributive Property
In mathematics, the distributive property is a crucial concept that allows us to simplify expressions. It states that when you multiply a single term by two or more terms inside a parenthesis, you distribute the multiplication over addition or subtraction.

For example, in the expression \( \sqrt{x}(\sqrt{x} - \sqrt{y}) \), the distributive property tells us to multiply \( \sqrt{x} \) by each term inside the parentheses separately.

Here鈥檚 how it works:
  • Multiply \( \sqrt{x} \) by \( \sqrt{x} \), resulting in \( \sqrt{x} \times \sqrt{x} \).
  • Then, multiply \( \sqrt{x} \) by \( -\sqrt{y} \), which gives \( -\sqrt{x} \times \sqrt{y} \).
This is how you distribute a term effectively across a subtraction or addition within parentheses.
Square Roots
A square root finds a number which, when multiplied by itself, gives the original number. The symbol for square root is \( \sqrt{} \).

When working with square roots, particularly when simplifying expressions involving them, some rules come in handy:
  • For any non-negative number \( a \), \( \sqrt{a^2} = a \).
  • The product of square roots can be simplified: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).
In the original exercise, \( \sqrt{x} \times \sqrt{x} \) simplifies to \( x \) because of the property \( \sqrt{a^2} = a \). Similarly, \( \sqrt{x} \times \sqrt{y} \) can be written as \( \sqrt{x \cdot y} \).

These simplifications help break down complex expressions into simpler forms.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. Usually, they represent a value and can be simplified or solved.

An algebraic expression like \( \sqrt{x}(\sqrt{x} - \sqrt{y}) \) combines variables under the square root sign with arithmetic operations. Each part of the expression should be addressed using algebra rules and properties, such as the distributive property and square root rules, to simplify the expression.

When dealing with algebraic expressions:
  • Identify terms and variables: look for common patterns or properties like \( x \) in the expression.
  • Use properties such as distributive and commutative to simplify.
  • Combine like terms to make expressions more straightforward.
Through these methods, algebraic expressions become more manageable, and you can solve or simplify them efficiently.

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