Problem 375
Question
In the following exercises, simplify by rationalizing the denominator. $$ \frac{15}{\sqrt{5}} $$
Step-by-Step Solution
Verified Answer
\frac{15}{\sqrt{5}} = 3 \sqrt{5}.
1Step 1: Identify the irrational part
Look at the denominator and see that it contains an irrational number, which is \( \sqrt{5} \).
2Step 2: Multiply by the conjugate
To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{5} \) to eliminate the square root. The fraction becomes \( \frac{15}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \).
3Step 3: Simplify the expression
Perform the multiplication in both the numerator and the denominator. \( \frac{15 \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{15 \sqrt{5}}{5} \).
4Step 4: Divide numerator by denominator
Simplify the fraction by dividing 15 by 5. \( \frac{15 \sqrt{5}}{5} = 3 \sqrt{5} \).
Key Concepts
Simplify FractionsRadicalsIrrational NumbersAlgebraic Expressions
Simplify Fractions
To simplify fractions, you aim to reduce the fraction to its simplest form. This often involves dividing both the numerator and denominator by their greatest common divisor (GCD).
In some problems, like the one given, simplifying involves more steps because the denominator includes a radical, necessitating rationalization.
Once you rationalize the denominator, further simplification can make the fraction easier to handle in calculations. Always look for ways to break down the numbers involved to their simplest forms.
In some problems, like the one given, simplifying involves more steps because the denominator includes a radical, necessitating rationalization.
Once you rationalize the denominator, further simplification can make the fraction easier to handle in calculations. Always look for ways to break down the numbers involved to their simplest forms.
Radicals
Radicals represent the roots of numbers, most commonly square roots. For example, \( \text{ √5} \) represents a number which, when multiplied by itself, equals 5.
Radicals can often appear in denominators, making calculations more complex.
To handle this, we often rationalize the denominator by eliminating the radical. This process involves multiplying both the numerator and the denominator by the same radical number. In our problem, this was \( \text{ √5} \). This helps to transform the fraction into a form that’s easier to work with and simplifies calculations.
Radicals can often appear in denominators, making calculations more complex.
To handle this, we often rationalize the denominator by eliminating the radical. This process involves multiplying both the numerator and the denominator by the same radical number. In our problem, this was \( \text{ √5} \). This helps to transform the fraction into a form that’s easier to work with and simplifies calculations.
Irrational Numbers
Irrational numbers are those which cannot be expressed as a simple fraction, like π (pi) and √5 (square root of 5). These numbers have decimal expansions that neither terminate nor repeat.
To manage them effectively in algebra problems, we often need to manipulate them through techniques like rationalizing the denominator.
By rationalizing, we transform a fraction containing an irrational number into one that is easier for both computation and comprehension. This is a critical step for simplifying expressions involving irrational numbers.
To manage them effectively in algebra problems, we often need to manipulate them through techniques like rationalizing the denominator.
By rationalizing, we transform a fraction containing an irrational number into one that is easier for both computation and comprehension. This is a critical step for simplifying expressions involving irrational numbers.
Algebraic Expressions
Algebraic expressions involve numbers, variables, and their operations (such as addition, subtraction, multiplication, and division). Simplifying these expressions can often involve handling fractions and radicals.
Consider the given exercise, where we simplify \( \frac{15}{\text{ √5}} \) by rationalizing the denominator. Key steps include identifying the parts of the expression, performing multiplications, and further simplification.
By mastering these techniques, solving and simplifying algebraic expressions becomes much more approachable.
This understanding is vital for pursuing advanced algebra and calculus.
Consider the given exercise, where we simplify \( \frac{15}{\text{ √5}} \) by rationalizing the denominator. Key steps include identifying the parts of the expression, performing multiplications, and further simplification.
By mastering these techniques, solving and simplifying algebraic expressions becomes much more approachable.
This understanding is vital for pursuing advanced algebra and calculus.
Other exercises in this chapter
Problem 373
In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{150 x^{2} y^{6}}}{\sqrt{6 x^{4} y^{2}}} $$
View solution Problem 374
In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{80 p^{3} q}}{\sqrt{5 p q^{5}}} $$
View solution Problem 376
In the following exercises, simplify by rationalizing the denominator. $$ \frac{3}{5 \sqrt{8}} $$
View solution Problem 377
In the following exercises, simplify by rationalizing the denominator. $$ \sqrt{\frac{8}{54}} $$
View solution