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Chapter 8: Problem 50

Find each product and simplify. $$ \sqrt{30} \cdot \sqrt{24} $$

Short Answer

Expert verified
\( \sqrt{30} \cdot \sqrt{24} = 12 \sqrt{5} \)

Step by step solution

01

- Multiply the Radicals

First, recognize that both numbers are under the radicals. You can multiply the radicals together under a single square root: ewline \ \[ \sqrt{30} \cdot \sqrt{24} = \sqrt{30 \cdot 24} \ \]
02

- Multiply the Numbers Under the Radical

Next, multiply the numbers 30 and 24 together: ewline \ \[ 30 \cdot 24 = 720 \ \] Now, we have: ewline \ \[ \sqrt{30 \cdot 24} = \sqrt{720} \ \]
03

- Simplify the Radical

To simplify \( \sqrt{720} \), first factor 720 into its prime factors:ewline \ \[ 720 = 2^4 \cdot 3^2 \cdot 5 \ \] Now express \( \sqrt{720} \) using these prime factors: ewline \ \[ \sqrt{720} = \sqrt{2^4 \cdot 3^2 \cdot 5} \ \] Then separate the factors under the square root: ewline \ \[ \sqrt{2^4} \cdot \sqrt{3^2} \cdot \sqrt{5} = 2^2 \cdot 3 \cdot \sqrt{5} = 4 \cdot 3 \cdot \sqrt{5} = 12\sqrt{5} \ \]

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Key Concepts

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

Multiplying Radicals
When multiplying radicals, think of it as combining numbers under a single square root. Suppose we need to multiply two radicals \(\root30\root24\). Start by merging them under one large square root: \[ \root30\root24=\root\bigg(30 \times 24\bigg)\bigg.\]. This simplifies our calculation by creating one larger number to work with. This step is important because it sets the stage for easier simplifications later on. Remember, the rule is \(\root a \times b=\root(ab)\), which holds for all radicals.
Prime Factorization
Prime factorization is breaking down a number into its simplest building blocks, which are prime numbers. For instance, to work with \(\root{720}\), we first turn 720 into its prime components: \[720=24 \times 32 \times 5.\] This can be further broken down as \[720=2^4 \times 3^2 \times 5.\]. Now, we have deconstructed 720 into only primes, which makes it easier to perform operations like taking square roots. Breaking down numbers into primes helps reveal their true structure, allowing for more straightforward calculations.
Simplifying Radicals
To simplify radicals, especially after getting a big number, we start with separating the prime factors under the square root. For example, simplifying \(\root{720}\) would go as follows: \[720=2^4 \times 3^2 \times 5\]. Then, apply the square root to each factor separately: \[ \root720= \root {2^4} \times \root {3^2} \times \root 5.\] This simplifies to \[2^2 \times 3 \times \root 5= 4 \times 3 \times \root 5=12 \root 5.\] The result, \(\root{720}\), is simplified to \(12 \root 5).\) By breaking down the radicals step-by-step, we can condense complex expressions into simpler forms.

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Most popular questions from this chapter

Simplify each radical expression. $$ \sqrt{\frac{8}{3}} \cdot \sqrt{\frac{512}{27}} $$

Solve each equation. (Hint: In Exercises 67 and 68, extend the concepts to fourth root radicals.) $$ \sqrt[4]{x^{2}+24 x}=3 $$

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{20 r^{10}} $$

Simplify each radical expression. $$ \sqrt{\frac{4}{50}} $$

Perform each operation and express the answer in simplest form. $$ 6 \sqrt[4]{9}(2 \sqrt[4]{9}-\sqrt[4]{27}) $$
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