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Chapter 10: Problem 56

Perform the indicated operations. Assume that all variables represent positive real numbers. $$ 5 \sqrt{\frac{288}{25}}+21 \frac{\sqrt{2}}{\sqrt{18}} $$

Short Answer

Expert verified
12 \sqrt{2} + 7.

Step by step solution

01

- Simplify the First Term

Start by simplifying the first term: \( 5 \sqrt{\frac{288}{25}} \). First, simplify the fraction under the square root: \( \sqrt{\frac{288}{25}} = \frac{\sqrt{288}}{\sqrt{25}} \). Since \( \sqrt{25} = 5 \), the expression becomes: \( \frac{\sqrt{288}}{5} \). Therefore, the term simplifies to: \( 5 \cdot \frac{\sqrt{288}}{5} = \sqrt{288} \).
02

- Simplify \( \sqrt{288} \)

Next, simplify \( \sqrt{288} \) by finding its prime factorization: \( 288 = 2^5 \cdot 3^2 \). So, \( \sqrt{288} = \sqrt{2^5 \cdot 3^2} = 2^{\frac{5}{2}} \cdot 3 = 4\sqrt{2} \cdot 3 = 12\sqrt{2} \).
03

- Simplify the Second Term

Now, simplify the second term: \( 21 \frac{\sqrt{2}}{\sqrt{18}} \). Simplify the fraction under the square root: \( \frac{\sqrt{2}}{\sqrt{18}} = \sqrt{\frac{2}{18}} = \sqrt{\frac{1}{9}} = \frac{1}{3} \). Therefore, the term simplifies to: \( 21 \cdot \frac{1}{3} = 7 \).
04

- Combine Terms

Combine the simplified terms from Steps 2 and 3: \( 12\sqrt{2} + 7 \). The final simplified expression is: \( 12 \sqrt{2} + 7 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying radicals
Simplifying radicals involves rewriting a radical expression, such as a square root, in its simplest form. The aim is to make the expression easier to work with while retaining its value. To simplify a radical, one can factorize the number inside the radical into its prime factors and then apply the product rule of square roots: \[ \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \]. \ For instance, to simplify \( \sqrt{288} \), you start by finding its prime factorization. \ **Step-by-Step Simplification:** \
    \
  • Factorize 288: \( 288 = 2^5 \cdot 3^2 \)
    • \
  • Apply the product rule: \( \sqrt{288} = \sqrt{2^5 \cdot 3^2} \)
    • \
  • Break it down: \( \sqrt{2^5} \cdot \sqrt{3^2} \)
    • \
  • Simplify further: \( \sqrt{2^4 \cdot 2} \cdot 3 = 2^2 \cdot \sqrt{2} \cdot 3 = 4 \sqrt{2} \cdot 3 \)
    • \
  • Combine: \(12 \sqrt{2} \)
    • \
These steps help in breaking down and simplifying the radical expressions effectively.
prime factorization
Prime factorization is breaking down a composite number into a product of its prime factors. This technique is vital when working with radicals and fractions to simplify expressions. Primes are numbers greater than 1 that have only two factors: 1 and themselves. \ **Example:** To find the prime factorization of 288: \
    \
  • Divide by the smallest prime (2): \( 288 \div 2 = 144 \)
    • \
  • Repeat division: \( 144 \div 2 = 72 \)
    • \
  • Continue until non-divisible: \( 72 \div 2 = 36 \), \( 36 \div 2 = 18 \), \( 18 \div 2 = 9 \).
    • \
  • Switch primes when needed: \( 9 \div 3 = 3 \), \( 3 \div 3 = 1 \).
    • \
\Hence, \( 288 = 2^5 \cdot 3^2 \). This prime factorization aids in further calculations, like simplifying radicals.
combining like terms
Combining like terms is a fundamental concept in algebra that involves simplifying expressions by merging terms that have the same variables raised to the same power. This simplifies algebraic operations and solutions. \ **Steps to Combine Like Terms:** \
    \
  • Identify like terms: Terms that have the same variable part (e.g., \( 3x \) and \( 5x \) are like terms, while \( 2x \) and \( 3y \) are not).
    • \
  • Use arithmetic: Combine the coefficients of like terms by addition or subtraction.
    • \
\ **Example in the Exercise:** Consider \( 12\sqrt{2} + 7 \). Here, there are no other \( \sqrt{2} \) terms or whole numbers to combine with \( 12\sqrt{2} \) and 7, respectively. Hence, \( 12\sqrt{2} + 7 \) remains as it is.

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