Problem 80 - Algebra 1

Question

Simplify the radical expression. $$\frac{1}{4} \sqrt{112}$$

Step-by-Step Solution

Answer

The simplified form of the given radical expression \( \frac{1}{4} \sqrt{112} \) is \( \sqrt{7} \).

Step 1: Find Prime Factors of the Radical's Argument

Firstly, find the prime factorization of 112. This can be done via division starting from 2. The prime factorization is: \( 2^4 * 7 \)

Step 2: Simplify the Radical

For each pair of same numbers, take one number out of the square root. Here, we get two pairs of 2, so we take two 2's out of the square root. The simplified square root will be: \( 2^2*\sqrt{7} = 4*\sqrt{7} \)

Step 3: Multiply by Coefficient

Multiply the simplified radical from step 2 by the coefficient \( \frac{1}{4} \). So, \( \frac{1}{4} * 4*\sqrt{7} = \sqrt{7} \)

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors, which are the prime numbers that multiply together to form the original number. For instance, when simplifying radical expressions, understanding the prime factorization of the number inside the radical (known as the radicand) is crucial. Prime numbers are the building blocks of all numbers and have only two distinct positive divisors: 1 and themselves.

When performing prime factorization, you typically start with the smallest prime number that divides the number, which is usually 2, then proceed to 3, 5, 7, and so on, only considering prime numbers. Each time you find a prime factor, you divide the number by that prime and continue the process with the quotient until you are left with 1.

Example of Prime Factorization

In our exercise, the radicand is 112. Beginning with 2, the smallest prime factor, you would divide 112 by 2 to get 56. Repeat this process by dividing 56 by 2 to get 28, 28 by 2 to get 14, and 14 by 2 to get 7, which is prime. Therefore, the prime factorization of 112 is \( 2^4 \times 7 \).

Radical Simplification

Radical simplification involves manipulations to turn a complex radical expression into its simplest form. To do this, one must know the essential rule that for any positive integers 'a' and 'b', and for any integer 'n', if both 'a' and 'b' are perfect nth powers, then \( \sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b} \). This concept is used to break down radicals into more manageable parts, specifically by separating the radicand into its prime factors. Also, if the index of the radical (which is 2 for a square root) is the same as the exponent of a prime factor, that factor can be taken out of the radical.

Using the prime factorization obtained earlier, we noted for 112 that \( 2^4 \) has a square, which allows two 2s to come out of the square root as a single 2. The simplified square root for \( 2^4\times7 \) thus becomes \( 2^2\times\sqrt{7} = 4\times\sqrt{7} \).

Coefficient Multiplication

Coefficient multiplication is a pervasive concept in algebra that involves multiplying a constant term, known as the coefficient, with a variable or an expression. In terms of radical simplification, it applies when we multiply the coefficient by the newly simplified form of the radical to obtain the final simplified expression. The coefficient in our exercise is \(\frac{1}{4}\), and the simplified radical is \(\sqrt{7}\) after taking out the \(2^2\) outside the radical sign as 4.

Carrying out the multiplication, we have \(\frac{1}{4} \times 4\times\sqrt{7}\), which simplifies to \(\sqrt{7}\) because \(\frac{1}{4} \times 4 = 1\) and multiplying by 1 does not change the value of the expression. This multiplication step is important as it combines the external coefficient with the simplified radical, resulting in the final answer.