Problem 15
Question
In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{x^{3}} \cdot \sqrt{4 x} $$
Step-by-Step Solution
Verified Answer
The simplest form of the product is \(2x^2\).
1Step 1: Simplify each square root individually
Start by simplifying each square root individually. The expression is \( \sqrt{x^3} \cdot \sqrt{4x} \).1. \( \sqrt{x^3} = \sqrt{x^2 \cdot x} = x \cdot \sqrt{x} \) because \( \sqrt{x^2} = x \).2. \( \sqrt{4x} = \sqrt{4} \cdot \sqrt{x} = 2\sqrt{x} \) because \( \sqrt{4} = 2 \).
2Step 2: Multiply the simplified expressions
Multiply the simplified square root expressions together:\( (x \cdot \sqrt{x}) \cdot (2 \cdot \sqrt{x}) = 2x \cdot \sqrt{x} \cdot \sqrt{x} \).Notice that you now have \( \sqrt{x} \cdot \sqrt{x} \) which can be simplified.
3Step 3: Simplify the product of square roots
Recall that \( \sqrt{x} \cdot \sqrt{x} = x \). Therefore, this expression simplifies further:\( 2x \cdot x = 2x^2 \).
4Step 4: Write the final product
Combine the results from previous steps:The product simplifies to \( 2x^2 \). Hence, \( \sqrt{x^3} \cdot \sqrt{4x} = 2x^2 \).
Key Concepts
Square RootsProperties of RadicalsAlgebraic Expressions
Square Roots
A square root finds a number which, when multiplied by itself, gives the original number. Let's take a look at the number 9. Its square root is 3 because when you multiply 3 by itself (3 x 3), you get 9. Similarly, the square root of 16 is 4 because 4 times 4 equals 16.
In algebra, finding the square root involves variables too. For instance, with the expression \( \sqrt{x^3} \), you can simplify it by recognizing that \( x^3 \) is \( x^2 \cdot x \). Since \( x^2 \) is a perfect square, its square root is x. This makes the expression \( x \cdot \sqrt{x} \). Using square roots is handy in simplifying expressions and solving equations. Whenever you see a square root, think about what number or expression, when squared, would return the radicand (the number inside the square root). Remember that not all numbers have a simple integer square root like 9 or 16. This is where approximations or radical form come into play.
In algebra, finding the square root involves variables too. For instance, with the expression \( \sqrt{x^3} \), you can simplify it by recognizing that \( x^3 \) is \( x^2 \cdot x \). Since \( x^2 \) is a perfect square, its square root is x. This makes the expression \( x \cdot \sqrt{x} \). Using square roots is handy in simplifying expressions and solving equations. Whenever you see a square root, think about what number or expression, when squared, would return the radicand (the number inside the square root). Remember that not all numbers have a simple integer square root like 9 or 16. This is where approximations or radical form come into play.
Properties of Radicals
Understanding the properties of radicals can simplify complex expressions significantly. One crucial property is that the square root of a product equals the product of the square roots. For example, \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \). This property helps when dealing with radical expressions consisting of multiple terms.
Another essential property is that \( \sqrt{a} \cdot \sqrt{a} = a \). This means if you multiply the square root of a number by itself, you get the original number. This was used to simplify our example expression from \( \sqrt{x} \cdot \sqrt{x} \) to x.
Moreover, radicals can sometimes be simplified by recognizing perfect squares. For instance, \( \sqrt{4} = 2 \) because 4 is a perfect square. These properties, when combined effectively, help reduce complex expressions to simpler forms, making them easier to solve or evaluate.
Another essential property is that \( \sqrt{a} \cdot \sqrt{a} = a \). This means if you multiply the square root of a number by itself, you get the original number. This was used to simplify our example expression from \( \sqrt{x} \cdot \sqrt{x} \) to x.
Moreover, radicals can sometimes be simplified by recognizing perfect squares. For instance, \( \sqrt{4} = 2 \) because 4 is a perfect square. These properties, when combined effectively, help reduce complex expressions to simpler forms, making them easier to solve or evaluate.
Algebraic Expressions
Algebraic expressions involve variables, numbers, and arithmetic operations. In this context, understanding how to manipulate expressions is key to simplifying and solving them.
When dealing with expressions like \( \sqrt{x^3} \cdot \sqrt{4x} \), it's important to be comfortable breaking down the components. You can split the expression into simpler parts, as shown in the simplification process: \( \sqrt{x^3} \) became \( x \cdot \sqrt{x} \), and \( \sqrt{4x} \) became \( 2\sqrt{x} \).
This breakdown makes it easy to multiply or add expressions, as is often required in algebra. The properties and rules governing arithmetic operations on whole numbers apply similarly to variables. Thus, practicing handling expressions with variables prepares one for more complex algebraic tasks in mathematics.
When dealing with expressions like \( \sqrt{x^3} \cdot \sqrt{4x} \), it's important to be comfortable breaking down the components. You can split the expression into simpler parts, as shown in the simplification process: \( \sqrt{x^3} \) became \( x \cdot \sqrt{x} \), and \( \sqrt{4x} \) became \( 2\sqrt{x} \).
This breakdown makes it easy to multiply or add expressions, as is often required in algebra. The properties and rules governing arithmetic operations on whole numbers apply similarly to variables. Thus, practicing handling expressions with variables prepares one for more complex algebraic tasks in mathematics.
Other exercises in this chapter
Problem 14
In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator o
View solution Problem 14
In \(3-14,\) determine whether each of the numbers is rational or irrational. $$ 2 \frac{1}{3}+\sqrt{3} $$
View solution Problem 15
In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt{169} $$
View solution Problem 15
In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a f
View solution