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Chapter 9: Problem 16

Simplify the radical expression. $$\sqrt{432}$$

Short Answer

Expert verified
The simplified form of the radical expression \(\sqrt{432}\) is \(12\sqrt{3}\).

Step by step solution

01

Factorize the number

Start by factoring the number under the radical, which is 432. A large perfect square that goes into 432 is 144. Therefore, the number 432 can be written as \(144 \times 3\).
02

Simplify the radical

Now simplify the expression under the square root. The square root of 144 is 12, so \(\sqrt{144}=12\). The new expression becomes \(12\sqrt{3}\).
03

Final Answer

Therefore, the simplified radical form is \(12\sqrt{3}\).

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Key Concepts

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

Simplifying Radicals
When simplifying radicals, the goal is to express the radical in its most basic form. Radicals can sometimes look complicated, but simplifying them can often make both the problem and solution more understandable. Here are the steps you need to follow for simplifying radicals:
  • Identify perfect squares in the number under the radical.
  • Factor the number into its prime components to spot these perfect squares.
  • Take the square root of the perfect square and move it outside the radical sign.
For instance, in the exercise, the number 432 was simplified to 144 x 3. As 144 is a perfect square, it was taken outside the radical as 12, resulting in the simplified form of \(12\sqrt{3}\). Breaking down the radical into smaller parts often unveils the simplicity hiding in complex expressions.
Square Roots
Square roots are one of the most common forms of radicals and are essential in simplifying radical expressions. In mathematical terms, a square root of a number is a value that, when multiplied by itself, gives the original number. The symbol \(\sqrt{}\) denotes the square root.
  • For example, \(\sqrt{16}\) equals 4 because \(4 \times 4 = 16\).
  • Understanding how to identify and calculate square roots is crucial in simplifying radicals.
  • Even large numbers can have perfect squares, making the process simpler as shown in the exercise with \(\sqrt{144} = 12\).
When faced with a complex square root, break it down into components and look for perfect squares to extract and simplify the expression.
Factorization
Factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. This process is particularly helpful in simplifying radicals, as it helps to identify perfect squares easily.
  • Begin by dividing the number by the smallest prime number, preferably starting with 2. If it's not divisible, move to the next prime.
  • Continue the factorization until all components are prime numbers.
  • Check for factors that form perfect squares and use them to simplify the radical.
In the exercise, 432 was factorized into \(144 \times 3\). Since 144 is a perfect square, it was extracted from under the radical, streamlining the expression to \(12\sqrt{3}\). Effective factorization can greatly reduce the complexity of radical expressions, making them easier to handle.

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Most popular questions from this chapter

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