Problem 45 - Algebra 1

Question

Simplify the expression. $$-2 \sqrt{27} \cdot \sqrt{3}$$

Step-by-Step Solution

Answer

-18

Step 1: Break Down the Numbers into Prime Factors

Start with determining the prime factorization of the numbers inside the square root. For 27, the prime factors are 3*3*3, thus \(\sqrt{27}\) can be rewritten as \(\sqrt{3^3}\). So, the initial expression \(-2 \sqrt{27} \cdot \sqrt{3}\) can be rewritten as \(-2 \sqrt{3^3} \cdot \sqrt{3}\).

Step 2: Simplify the Multiplication of the Square Roots

According to the property of radicals, the multiplication of square roots can be written as the square root of the multiplication of the numbers. Therefore, replace \(\sqrt{3^3} \cdot \sqrt{3}\) with \(\sqrt{3^3 \cdot 3}\). The expression becomes \(-2 \sqrt{3^3 \cdot 3}\).

Step 3: Combine like-terms Inside the Square Root

Now, multiply the numbers inside the square root \(\sqrt{3^3 \cdot 3}\) to get \(\sqrt{3^4}\). So, the expression becomes \(-2 \sqrt{3^4}\).

Step 4: Simplify Further

Simplify \(\sqrt{3^4}\) to \(3^2\) which equals to 9. Our final expression is now \(-2 * 9\).

Step 5: Simplify the Final Expression

Multiply -2 by 9 to get -18.

Key Concepts

Prime Factorization

Prime factorization is a method of breaking down a composite number into its prime factors, which are numbers that are only divisible by 1 and themselves. For instance, when simplifying square roots, prime factorization is an essential first step to determine which numbers can be taken out of the radical.

Let's consider the given exercise \( -2 \sqrt{27} \cdot \sqrt{3} \). The number 27 can be factorized into prime factors as 3 \( \times \) 3 \( \times \) 3, or \(3^3\). The trick here is to look for pairs of prime factors, since a pair of primes under a square root can be simplified to a single number outside the square root. This process is particularly useful when dealing with larger numbers as it systematically breaks down the problem into smaller and more manageable parts.

Properties of Radicals

Simplification of Square Roots

The properties of radicals are rules that define how radical expressions can be manipulated and simplified. One of the key properties is that the square root of a product can be expressed as the product of the square roots. This implies that \( \sqrt{a \cdot b} \) is the same as \( \sqrt{a} \cdot \sqrt{b} \).

In the context of our example, this property allows us to combine \( \sqrt{3^3} \) and \( \sqrt{3} \) into \( \sqrt{3^3 \cdot 3} \) because \( \sqrt{27} \) is equivalent to \( \sqrt{3^3} \) and \( \sqrt{3} \) remains unchanged. By understanding and applying these properties, students can simplify radical expressions and pave the way for further operations, such as combining like terms.

Combining Like Terms

Merging Similar Factors Under the Radical

Combining like terms within radicals involves merging similar factors to reduce the expression to a simpler form. In our exercise, we have a radical expression \(-2 \sqrt{3^3} \cdot \sqrt{3}\). Upon following the properties of radicals, we combine the factors inside the square root to obtain \(-2 \sqrt{3^4}\).

This process for radicals is analogous to combining like terms in algebraic expressions. Here, the 'like terms' are actually identical factors within the radical that can be multiplied together. Once these are combined, \(3^4\) simplifies to a square number, \(9\), which can be easily taken out of the square root, drastically decreasing the complexity of the problem. Understanding how to efficiently combine like terms within radicals can significantly simplify solving and simplifying mathematical expressions.