Problem 21 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{24 x^{4}} $$

Step-by-Step Solution

Answer

The simplified radical expression is 2√(6)x^2.

Step 1: Find the prime factors of the radicand

To simplify the radical expression, first, find the prime factors of the coefficient (the number part) and simplify the variable part of the expression (x^4). The prime factorization of 24 is \(2 * 2 * 2 * 3\) or \(2^3 * 3\). The simplified expression inside the square root becomes: $$ \sqrt{2^3 * 3 * x^4} $$

Step 2: Simplify the exponents

The expression inside the square root has exponents of 3 for the number 2 and 4 for the variable x. Use the property of square roots, \(\sqrt{a^m * b^n} = a^{m/2} * b^{n/2}\), to simplify the expression. For this problem, m = 3 for the number 2 and n = 4 for the variable x: $$ \sqrt{2^3 * 3 * x^4} = 2^{\frac{3}{2}} * 3^{\frac{1}{2}} * x^{\frac{4}{2}} $$

Step 3: Evaluate the simplified exponents

Evaluate each part of the expression with a simplified exponent. For the number 2 (with an exponent of 3/2), the square root of \(2^3 = 2 * 2 * 2 = 8\), which gives \(2\sqrt{2}\). For the number 3 (with an exponent of 1/2), it is just \(\sqrt{3}\). And finally, for the variable x (with an exponent of 4/2), we get \(x^{4/2} = x^2\). So, the simplified expression is: $$ 2\sqrt{2} * \sqrt{3} * x^2 $$

Step 4: Combine the remaining parts

Combine the square roots by multiplying them and then multiply the result by the variable part x^2: $$ (2\sqrt{2} * \sqrt{3}) * x^2 = 2\sqrt{6} * x^2 $$

Step 5: Final Answer

The simplified radical expression for the given problem is: $$ \sqrt{24 x^{4}} = 2\sqrt{6}x^2 $$

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its smallest prime numbers that can be multiplied together to get the original number. It's like finding the 'building blocks' of a number.

• For example, the number 24 can be broken down as follows: Start with the smallest prime number 2, and divide 24 by 2 repeatedly until you can't divide evenly anymore.
• You get: 24 ÷ 2 = 12 -> 12 ÷ 2 = 6 -> 6 ÷ 2 = 3.
• The prime numbers we are left with are 2, 2, 2, and 3, which means 24 = 2 × 2 × 2 × 3 or written in exponent form, 24 = 2^3 × 3.

This method is very useful for simplifying radicals, as it helps in identifying perfect squares that can be easily extracted from under a square root symbol.

Exponents

Exponents are a way to represent repeated multiplication of the same factor. They are written as a small number to the upper right of a base number.

• For example, 3^2 means 3 multiplied by itself, which is 3 × 3 = 9.
• In the expression \(2^3\), the exponent 3 tells us to multiply 2 by itself three times: 2 × 2 × 2 = 8.

Understanding exponents is especially important in simplifying expressions involving square roots. For instance, when simplifying \(x^4\) in a square root, we divide the exponent by 2 due to the property of square roots \(\sqrt{x^n} = x^{n/2}\). This helps in simplifying radical expressions effectively and is a fundamental skill in algebra.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. It's represented with the radical symbol \(\sqrt{}\). There's a special relationship between squares and square roots:

• If \(x = y^2\), then \(y = \sqrt{x}\).
• For example, the square root of 9 is 3 because 3 × 3 = 9.

When simplifying expressions with square roots, it's crucial to handle both numerical and variable parts properly. For instance, in the expression \(\sqrt{24x^4}\):

• We simplify \(24\) by breaking it down to prime factors, allowing easy extraction of perfect squares, like \(2^2\) giving \(2\) out of the square root.
• The variable part \(x^4\) becomes \(x^2\) when simplified because \(\sqrt{x^4} = x^{4/2} = x^2\).

This step is key to simplifying complex radical expressions, often leaving you with an easier expression to work with.