Problem 52
Question
Rationalize each denominator and simplify if possible. See Section 10.5. $$ \frac{10}{\sqrt{5}} $$
Step-by-Step Solution
Verified Answer
The rationalized and simplified form is \(2\sqrt{5}\).
1Step 1: Identify the Radical in the Denominator
Notice that the denominator of the fraction \( \frac{10}{\sqrt{5}} \) is \( \sqrt{5} \). Our goal is to remove this radical from the denominator by rationalizing it.
2Step 2: Multiply Numerator and Denominator by the Conjugate
To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{5} \). This is because the product of a radical and itself removes the radical: \( \sqrt{5} \times \sqrt{5} = 5 \). This gives us \( \frac{10 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} \).
3Step 3: Simplify the Expression
Simplify the expression \( \frac{10 \times \sqrt{5}}{5} \). The denominator becomes \( \sqrt{5} \times \sqrt{5} = 5 \). The expression is simplified to \( \frac{10\sqrt{5}}{5} = 2\sqrt{5} \), because \( \frac{10}{5} = 2 \).
Key Concepts
Understanding RadicalsSimplification of ExpressionsUtilizing ConjugatesExploring Algebraic Expressions
Understanding Radicals
A radical is a mathematical symbol that is used to represent the root of a number. The most common radical is the square root, which is represented by the symbol \( \sqrt{} \). When you see \( \sqrt{5} \), it means you are looking for a number that, when multiplied by itself, gives you 5. Radicals play a significant role in mathematics and are particularly useful when dealing with non-perfect squares.
It's important to understand that radicals can often make manipulation of algebraic expressions in equations cumbersome, especially when they appear in the denominator. That's why mathematicians often prefer to rationalize the denominator, or in other words, find an equivalent expression without radicals in the denominator.
It's important to understand that radicals can often make manipulation of algebraic expressions in equations cumbersome, especially when they appear in the denominator. That's why mathematicians often prefer to rationalize the denominator, or in other words, find an equivalent expression without radicals in the denominator.
Simplification of Expressions
Simplification in mathematics involves reducing an expression to its simplest form. This process ensures that the expression is easier to work with. When simplifying fractions, you aim to perform operations on both the numerator and the denominator until you can't simplify any further.
In the example \( \frac{10}{\sqrt{5}} \), simplification involves multiplying both the numerator and the denominator by \( \sqrt{5} \) to remove the radical from the denominator. The actions result in \( \frac{10\sqrt{5}}{5} \), and further reduction gives \( 2\sqrt{5} \). Thus, simplification eloquently transforms complex expressions into more user-friendly forms, creating expressions that are both interpretable and operable.
In the example \( \frac{10}{\sqrt{5}} \), simplification involves multiplying both the numerator and the denominator by \( \sqrt{5} \) to remove the radical from the denominator. The actions result in \( \frac{10\sqrt{5}}{5} \), and further reduction gives \( 2\sqrt{5} \). Thus, simplification eloquently transforms complex expressions into more user-friendly forms, creating expressions that are both interpretable and operable.
Utilizing Conjugates
A conjugate refers to changing the sign between two terms in a binomial. With rationalization, especially when dealing with binomials involving radicals, the conjugate plays a remarkable role. In earlier grades, you might have been taught about products of conjugate pairs: products like \((a+b)(a-b) = a^2 - b^2\).
In this specific exercise with \( \frac{10}{\sqrt{5}} \), the term "conjugate" might be a bit misleading since only a single radical is present. Thus, you multiply by the radical itself to clear it from the denominator. However, understanding the concept of conjugates becomes especially handy when you have more complex denominators with additional terms, such as \( \sqrt{a} - b \). The conjugate would then be \( \sqrt{a} + b \). Multiplying by this form efficiently eliminates radicals from denominator settings and prevents potential cancellation mishaps.
In this specific exercise with \( \frac{10}{\sqrt{5}} \), the term "conjugate" might be a bit misleading since only a single radical is present. Thus, you multiply by the radical itself to clear it from the denominator. However, understanding the concept of conjugates becomes especially handy when you have more complex denominators with additional terms, such as \( \sqrt{a} - b \). The conjugate would then be \( \sqrt{a} + b \). Multiplying by this form efficiently eliminates radicals from denominator settings and prevents potential cancellation mishaps.
Exploring Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and operators representing a value. Simplifying such expressions forms a cornerstone of algebra, especially when radicals and fractions are involved.
Rationalizing denominators by multiplying by a radical or its conjugate is an integral process in algebraic simplification because it allows expressions to be more manageable and operational, thus making them easier to use in subsequent calculations and problem-solving.
Rationalizing denominators by multiplying by a radical or its conjugate is an integral process in algebraic simplification because it allows expressions to be more manageable and operational, thus making them easier to use in subsequent calculations and problem-solving.
- Expressions like \( \frac{10}{\sqrt{5}} \) transform easily into a cleaner form \( 2\sqrt{5} \).
- Working with algebraic expressions ensures problem-solving becomes approachable and logical.
- This application of operations reinforces understanding and manipulation of expressions in multiple scenarios of mathematics.
Other exercises in this chapter
Problem 51
Rationalize each denominator and simplify if possible. See Section 10.5. $$ \frac{4 \sqrt{7}}{\sqrt{6}} $$
View solution Problem 52
The graph of each equation is an ellipse. Determine which distance is longer, the distance between the \(x\) -intercepts or the distance between the y-intercept
View solution Problem 53
If you are given a list of equations of circles, parabolas, ellipses, and hyperbolas, explain how you could distinguish the different conic sections from their
View solution Problem 54
We know that \(x^{2}+y^{2}=25\) is the equation of a circle. Rewrite the equation so that the right side is equal to \(1 .\) Which type of conic section does th
View solution