Problem 27
Question
Simplify each expression. $$\sqrt{\frac{360}{361}}$$
Step-by-Step Solution
Verified Answer
\( \sqrt{\frac{360}{361}} = \frac{6\sqrt{10}}{19} \).
1Step 1: Understand the Expression
We need to simplify the expression \( \sqrt{\frac{360}{361}} \). This expression involves a square root of a fraction.
2Step 2: Simplified Inside Square Root
Recognize that the expression \( \sqrt{\frac{a}{b}} \) can be rewritten using the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Therefore, \( \sqrt{\frac{360}{361}} = \frac{\sqrt{360}}{\sqrt{361}} \).
3Step 3: Calculate the Denominator
Realize that \( \sqrt{361} \) is 19 because 361 is a perfect square (since \( 19 \times 19 = 361 \)). Thus, the expression becomes \( \frac{\sqrt{360}}{19} \).
4Step 4: Simplify the Numerator using Prime Factorization
Find the prime factorization of 360, which is \( 2^3 \times 3^2 \times 5 \). Therefore, \( \sqrt{360} = \sqrt{2^3 \times 3^2 \times 5} = \sqrt{2^2 \times 2 \times 3^2 \times 5} = 2 \times 3 \times \sqrt{2 \times 5} = 6\sqrt{10} \).
5Step 5: Simplify the Expression
Substitute back into the expression: \( \frac{\sqrt{360}}{19} \) becomes \( \frac{6\sqrt{10}}{19} \).
Key Concepts
Square rootsFraction simplificationPrime factorization
Square roots
Square roots are fascinating concepts in algebra, representing a number that, when multiplied by itself, yields the original number. For example, the square root of 361 is 19 because if you multiply 19 by itself, you get 361.
A useful property of square roots when dealing with fractions is that the square root of a fraction, such as \( \sqrt{\frac{a}{b}} \), can be split into two separate square roots: \( \frac{\sqrt{a}}{\sqrt{b}} \). This property makes simplifying complex expressions more manageable.
In this exercise, applying this property to \( \sqrt{\frac{360}{361}} \) helped to separate the task into smaller, more digestible parts, allowing for easier simplification. Recognizing that 361 is a perfect square simplifies the process because its square root is an integer.
A useful property of square roots when dealing with fractions is that the square root of a fraction, such as \( \sqrt{\frac{a}{b}} \), can be split into two separate square roots: \( \frac{\sqrt{a}}{\sqrt{b}} \). This property makes simplifying complex expressions more manageable.
In this exercise, applying this property to \( \sqrt{\frac{360}{361}} \) helped to separate the task into smaller, more digestible parts, allowing for easier simplification. Recognizing that 361 is a perfect square simplifies the process because its square root is an integer.
Fraction simplification
When simplifying a fraction, the goal is to make the fraction as simple as possible. Fractions represent a division between two numbers, known as the numerator and the denominator.
One common technique is to apply properties like breaking down a fraction's terms under a square root. By recognizing and calculating the square roots of both the numerator and denominator separately, complex expressions can be reduced to simpler forms.
One common technique is to apply properties like breaking down a fraction's terms under a square root. By recognizing and calculating the square roots of both the numerator and denominator separately, complex expressions can be reduced to simpler forms.
- This makes it easier to handle expressions mathematically by reducing them to integer components or simpler radicals.
- For example, \( \sqrt{\frac{360}{361}} \) simplifies to \( \frac{\sqrt{360}}{19} \), since 361 is a perfect square with 19 as its square root.
Prime factorization
Prime factorization involves breaking down a number into a set of prime numbers that multiply together to result in the original number. Prime numbers are integers greater than 1 that are only divisible by 1 and themselves, like 2, 3, 5, and 7.
This technique plays a crucial role in simplifying square roots, especially for numbers that are not perfect squares.
By finding the prime factors, we can rearrange the radicals to achieve a simpler form. Consider the number 360, which can be expressed as its prime factorization: \( 2^3 \times 3^2 \times 5 \). From this, \( \sqrt{360} \) can be broken down further as \( \sqrt{2^2 \times 2 \times 3^2 \times 5} \), allowing us to remove the perfect squares from the root, resulting in \( 6\sqrt{10} \).
This technique plays a crucial role in simplifying square roots, especially for numbers that are not perfect squares.
By finding the prime factors, we can rearrange the radicals to achieve a simpler form. Consider the number 360, which can be expressed as its prime factorization: \( 2^3 \times 3^2 \times 5 \). From this, \( \sqrt{360} \) can be broken down further as \( \sqrt{2^2 \times 2 \times 3^2 \times 5} \), allowing us to remove the perfect squares from the root, resulting in \( 6\sqrt{10} \).
- This aids in rationalizing expressions and dealing with more complicated algebraic equations.
- Understanding this concept helps visualize where complex expressions can be simplified.
Other exercises in this chapter
Problem 27
For the following exercises, expand the binomial. $$(4 p+9)^{2}$$
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Divide the rational expressions. $$ \frac{18 d^{2}+77 d-18}{27 d^{2}-15 d+2} \div \frac{3 d^{2}+29 d-44}{9 d^{2}-15 d+4} $$
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For the following exercises, simplify the given expression. Write answers with positive exponents. $$ \left(b^{3} c^{4}\right)^{2} $$
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Simplify the given expression. $$ 12(3-1) \div 6 $$
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