Problem 62
Question
Simplify. $$ \sqrt{288} $$
Step-by-Step Solution
Verified Answer
\(12 \sqrt{2}\).
1Step 1 - Identify Prime Factors
First, factorize 288 into its prime factors. \(288 = 2^5 \times 3^2\).
2Step 2 - Apply Square Root to Prime Factors
The square root of a product is the product of the square roots, so apply the square root to the prime factors. \(\sqrt{288} = \sqrt{2^5 \times 3^2}\).
3Step 3 - Simplify Each Term
Simplify \(\sqrt{2^5}\) and \(\sqrt{3^2}\). \(\sqrt{2^5} = \sqrt{2^4 \times 2} = 2^2 \sqrt{2} = 4 \sqrt{2}\). Similarly, \(\sqrt{3^2} = 3\).
4Step 4 - Combine Simplified Terms
Combine the simplified terms. \(\sqrt{288} = 4 \sqrt{2} \times 3 = 12 \sqrt{2}\).
Key Concepts
Prime FactorizationSquare RootsExponentsSimplification
Prime Factorization
Prime factorization is an essential step in simplifying square roots. It involves breaking down a composite number into its prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. For example, to factorize 288, we keep dividing it by prime numbers until we are left with only prime numbers:
As a result, the prime factorization of 288 is written as: \( 288 = 2^5 \times 3^2 \)
Knowing how to find prime factors helps make the simplification of square roots easier.
- 288 ÷ 2 = 144
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
As a result, the prime factorization of 288 is written as: \( 288 = 2^5 \times 3^2 \)
Knowing how to find prime factors helps make the simplification of square roots easier.
Square Roots
Square roots are a fundamental concept in algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. For a radical like \( \sqrt{288} \) we can use its prime factorization to simplify it.
When finding the square root of a product of numbers, we can apply the square root to each factor separately:
\[ \sqrt{288} = \sqrt{2^5 \times 3^2} \]
This makes it easier to simplify complex square roots into more manageable terms.
When finding the square root of a product of numbers, we can apply the square root to each factor separately:
\[ \sqrt{288} = \sqrt{2^5 \times 3^2} \]
This makes it easier to simplify complex square roots into more manageable terms.
Exponents
Exponents are used to express repeated multiplication of the same number. For example, \( 2^5 \) represents \( 2 \) multiplied by itself 5 times: \( 2 \times 2 \times 2\times 2\times 2 = 32 \).
When dealing with square roots, exponents help simplify the process. Consider \( 2^5 \). We can express it as \( 2^4 \times 2 \):
This approach makes it easier to combine the results and simplify the original square root.
When dealing with square roots, exponents help simplify the process. Consider \( 2^5 \). We can express it as \( 2^4 \times 2 \):
- \( 2^4 \) is a perfect square and simplifies to \( (2^2)^2 = 4^2 \), so \( \sqrt{2^4} = 2^2 = 4 \).
- \( \sqrt{3^2} = 3 \) as \(3^2 \) is already a perfect square.
This approach makes it easier to combine the results and simplify the original square root.
Simplification
Simplification refers to expressing a mathematical expression in its simplest form. For square roots, it involves breaking down the terms and combining like terms.
Using our example, we found that \( \sqrt{2^5} = 4 \sqrt{2} \) and \( \sqrt{3^2} = 3 \).
Combining these terms, we get:
Using our example, we found that \( \sqrt{2^5} = 4 \sqrt{2} \) and \( \sqrt{3^2} = 3 \).
Combining these terms, we get:
- \( \sqrt{288} = 4 \sqrt{2} \times 3 = 12 \sqrt{2} \).