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Chapter 10: Problem 55

Simplify. See Examples 3 and 4 $$ \sqrt{9 x^{7} y^{9}} $$

Short Answer

Expert verified
The simplified expression is \(3x^3y^4\sqrt{xy}\).

Step by step solution

01

Factor Under the Square Root

Factor the expression inside the square root to check for any perfect squares. Note that 9 is a perfect square, and we can simplify the powers of variables as well: \[ 9 x^7 y^9 = (3^2) (x^6 imes x) (y^8 imes y) \] where \(x^6\) and \(y^8\) are perfect squares.
02

Simplify the Square Root

Apply the property \(\sqrt{a^2} = a\) to each perfect square factor identified. Simplify inside the square root: \[ \sqrt{3^2} = 3, \quad \sqrt{x^6} = x^3, \quad \text{and} \quad \sqrt{y^8} = y^4 \]Now, let's write the expression as: \[ \sqrt{9x^6y^8} \times \sqrt{xy} = 3x^3y^4 \sqrt{xy} \]
03

Final Expression

Combine the simplified terms to get the final expression. Thus, the expression inside the radical is fully simplified:\[ 3x^3y^4\sqrt{xy} \]

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Key Concepts

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

Perfect Squares
When we talk about perfect squares, we refer to numbers or expressions that can be written as the square of another whole number or entire expression. For example, the number 9 is a perfect square because it equals \( 3^2 \).
Similarly, when working with algebraic expressions, like \( x^6 \) or \( y^8 \), they can also be considered perfect squares.
Let's break it down:
  • If you take \( x^6 \), this can be simplified to \((x^3)^2\), showing that it is a perfect square.
  • Similarly, \( y^8 \) can be written as \((y^4)^2\), also qualifying as a perfect square.
Identifying perfect squares within an equation or expression is crucial for simplifying radicals. This is the foundation for recognizing factors that can be simplified using square root properties.
Square Root Properties
Square root properties make it easier to work with radicals. The key property is \( \sqrt{a^2} = a \), which allows us to extract numbers or expressions from under the square root.
Using this property, you can see how simplifying radicals becomes more straightforward. For example:
  • \( \sqrt{9} = 3 \) because 9 is \( 3^2 \).
  • For variables, \( \sqrt{x^6} = x^3 \), and \( \sqrt{y^8} = y^4 \).
By applying this property, we convert the square root of perfect square terms into simpler expressions.
This simplification helps in solving equations and making calculations more manageable.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They can involve many different components including constants, coefficients, and powers of variables.
In the context of simplifying radicals, understanding how to manipulate these expressions is essential.
For instance, in the expression \( \sqrt{9 x^7 y^9} \), it combines both numeric and variable components. Here鈥檚 a breakdown of the simplification:
  • Factor numeric components, like the perfect square 9, separately from the variables.
  • Handle the variables by breaking down the powers into factors that form perfect squares, such as \( x^6 \) and \( y^8 \).
By applying the properties of perfect squares and square roots, you streamline these complex expressions, simplifying them to a form that is much easier to work with.

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Most popular questions from this chapter

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as \(8 \cdot 3,\) because 8 is a perfect cube. $$ 80 $$

Identify the domain and then graph each function. $$f(x)=\sqrt{x-3}$$ use the following table. $$\begin{array}{|c|c|} \hline x & {f(x)} \\ \hline 3 & {} \\ \hline 4 & {} \\ \hline 7 \\ \hline 12 & {} \\ \hline \end{array}$$

Fill in each box with the correct expression $$ \frac{\square}{x^{-2 / 5}}=x^{3 / 5} $$

Use rational expressions to write as a single radical expression. $$ \sqrt{5 r} \cdot \sqrt[3]{s} $$

Factor the common factor from the given expression. $$ x^{2 / 7} ; x^{3 / 7}-2 x^{2 / 7} $$
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