Problem 27
Question
Use the quotient rule to simplify the expressions in Exercises \(23-32\) Assume that \(x>0\) $$\frac{\sqrt{48 x^{3}}}{\sqrt{3 x}}$$
Step-by-Step Solution
Verified Answer
The simplified form of the expression \(\frac{\sqrt{48 x^{3}}}{\sqrt{3 x}}\) is \(4x\)
1Step 1: Apply the Quotient Rule
Apply the quotient rule to simplify \(\frac{\sqrt{48 x^{3}}}{\sqrt{3 x}}\) to \(\sqrt{\frac{48 x^{3}}{3 x}}\)
2Step 2: Simplify the Fraction
Simplify the fraction within the square root, \(\frac{48 x^{3}}{3 x}\), by canceling the common factors to get \(\frac{16 x^{2}}{1}\), or simply, 16x^2
3Step 3: Simplfy the square root
On simplifying the square root, we find that \(\sqrt{16 x^{2}}\) is equal to \(4x\)
Key Concepts
Simplifying expressionsSquare rootsFraction simplificationAlgebraic expressions
Simplifying expressions
Simplifying mathematical expressions is a fundamental skill in algebra. It involves reducing a complex expression into its simplest form.
This process makes it easier to work with and understand.To simplify, first identify similar terms, constants, and variables in the expression.
Combine like terms to reduce the complexity of the expression. For example in our exercise, the expression \(\frac{\sqrt{48 x^{3}}}{\sqrt{3 x}}\) was initially complex.By applying rules such as the quotient rule we can simplify and transform an expression into an equivalent but simpler one.
This will allow us to further manipulate and solve mathematical problems. The goal is to make the expression as easy as possible to evaluate or interpret.
This process makes it easier to work with and understand.To simplify, first identify similar terms, constants, and variables in the expression.
Combine like terms to reduce the complexity of the expression. For example in our exercise, the expression \(\frac{\sqrt{48 x^{3}}}{\sqrt{3 x}}\) was initially complex.By applying rules such as the quotient rule we can simplify and transform an expression into an equivalent but simpler one.
This will allow us to further manipulate and solve mathematical problems. The goal is to make the expression as easy as possible to evaluate or interpret.
Square roots
The square root is a unique mathematical operation that determines what number, when multiplied by itself, will equal the original number.
Square roots are typically denoted by the radical symbol \(\sqrt{}\). For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\). When dealing with algebraic expressions under a square root, like our exercise \(\sqrt{48x^3}\), the rules are the same—break it down:
Square roots are typically denoted by the radical symbol \(\sqrt{}\). For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\). When dealing with algebraic expressions under a square root, like our exercise \(\sqrt{48x^3}\), the rules are the same—break it down:
- Find the square root of each term under the radical separately.
- In our case, simplify \(\sqrt{48}\) and \(\sqrt{x^3}\) independently.
Fraction simplification
Simplifying fractions involves reducing fractions to their simplest form.
This usually means dividing both the numerator and the denominator by their greatest common factor.In the context of our problem, we had the fraction \(\frac{48x^3}{3x}\) inside the square root.
By identifying common factors (here, the number 3 and the variable \(x\)), we simplify the fraction:
This usually means dividing both the numerator and the denominator by their greatest common factor.In the context of our problem, we had the fraction \(\frac{48x^3}{3x}\) inside the square root.
By identifying common factors (here, the number 3 and the variable \(x\)), we simplify the fraction:
- The number 48 divided by 3 gives 16.
- The variable \(x^3\) divided by \(x\) leaves \(x^2\).
Algebraic expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations.
These expressions are fundamental in algebra and require different approaches to simplify or solve them.When you encounter something like \(\sqrt{48x^3}\), it is an algebraic expression.
Simplifying such expressions involves a keen understanding of several mathematical rules and properties.Often, handling algebraic expressions involves:
These expressions are fundamental in algebra and require different approaches to simplify or solve them.When you encounter something like \(\sqrt{48x^3}\), it is an algebraic expression.
Simplifying such expressions involves a keen understanding of several mathematical rules and properties.Often, handling algebraic expressions involves:
- Recognizing and applying operations like addition, subtraction, multiplication, and division.
- Using power and root rules where necessary.
- Implementing techniques like factoring, expanding, or applying specific rules such as the quotient rule.
Other exercises in this chapter
Problem 27
Multiply or divide as indicated. $$\frac{4 x^{2}+10}{x-3} \div \frac{6 x^{2}+15}{x^{2}-9}$$
View solution Problem 27
Find each product. $$\left(5 x^{2}-4\right)\left(3 x^{2}-7\right)$$
View solution Problem 28
Find the intersection of the sets. $$\\{w, y, z\\} \cap \varnothing$$
View solution Problem 28
Simplify each exponential expression. $$x^{11} \cdot x^{5}$$
View solution