Problem 59 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \frac{2 \sqrt{32}}{\sqrt{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: \(\frac{2 \sqrt{32}}{\sqrt{3}}\). Answer: \(\frac{4\sqrt{6}}{3}\).

Step 1: Simplify the square roots

We have the expression: $$ \frac{2 \sqrt{32}}{\sqrt{3}} $$ Let's simplify the square root in the numerator, \(\sqrt{32}\). We can determine the prime factorization of 32 as \(2^5\), so we can rewrite the expression as: $$ \frac{2\sqrt{2^5}}{\sqrt{3}} $$ Since \(\sqrt{a^2} = a\) and from the property \(\sqrt{a^n} = a^{n/2}\), we can further rewrite the expression: $$ \frac{2(2^{\frac{5}{2}})}{\sqrt{3}} $$

Step 2: Simplify the coefficients

Using the exponent properties, we can simplify the expression further: $$ \frac{2(2^{\frac{5}{2}})}{\sqrt{3}} = \frac{2(2^{2 \times \frac{1}{2}} \times 2^{\frac{1}{2}})}{\sqrt{3}} $$ Since \(2^{\frac{1}{2}}\) is simply the square root of 2 and \(2^{2 \times \frac{1}{2}}\) is 2, the expression becomes: $$ \frac{2(2 \times 2^{\frac{1}{2}})}{\sqrt{3}} = \frac{4\sqrt{2}}{\sqrt{3}} $$

Step 3: Rationalize the denominator

In order to rationalize the denominator, we multiply both the numerator and the denominator by \(\sqrt{3}\): $$ \frac{4\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{6}}{3} $$

Step 4: Write the simplified expression

The simplified expression is: $$ \frac{4\sqrt{6}}{3} $$

Key Concepts

Prime Factorization

When simplifying radical expressions, prime factorization is a crucial step. It's the process of breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no other divisors besides 1 and themselves.

Let's consider the number 32 from our original expression \( \frac{2 \sqrt{32}}{\sqrt{3}} \). To find its prime factors, we divide 32 by 2 repeatedly, as 2 is a prime number, until we can no longer divide evenly:

• 32 divided by 2 equals 16,
• 16 divided by 2 equals 8,
• 8 divided by 2 equals 4,
• 4 divided by 2 equals 2,
• 2 divided by 2 equals 1.

The prime factorization of 32 is thus \( 2^5 \). This means we can rewrite \( \sqrt{32} \) as \( \sqrt{2^5} \). By understanding prime factorization, we can simplify \( \sqrt{32} \) in radical expressions, such as reducing \( \sqrt{32} \) to \( 2^{\frac{5}{2}} \), which facilitates simpler mathematical calculations.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers, like square roots, from the denominator of a fraction. This is a common procedure in mathematics to make expressions easier to work with. Let's examine how this applies:

Given the expression \( \frac{4\sqrt{2}}{\sqrt{3}} \), having \( \sqrt{3} \) in the denominator makes the fraction irrational. To rationalize it, we multiply both the numerator and the denominator by \( \sqrt{3} \). This keeps the value of the expression the same since we're effectively multiplying by 1 (\( \frac{\sqrt{3}}{\sqrt{3}} = 1 \)).

• Multiply the numerator: \( 4\sqrt{2} \times \sqrt{3} = 4\sqrt{6} \).
• Multiply the denominator: \( \sqrt{3} \times \sqrt{3} = 3 \).

So, the expression becomes \( \frac{4\sqrt{6}}{3} \). By rationalizing the denominator, we arrive at an expression that is more standard and often more practical for further calculations.

Properties of Exponents

The properties of exponents are fundamental tools, essential not just for simplifying radical expressions, but for algebra in general. Understanding these properties helps in expressing and simplifying numbers raised to powers. Here are some key exponent properties:

• Product of Powers: \( a^m \times a^n = a^{m+n} \). To multiply powers with the same base, add their exponents.
• Power of a Power: \( (a^m)^n = a^{m\times n} \). To raise a power to another power, multiply the exponents.
• Power of a Product: \( (ab)^n = a^n \times b^n \). Each factor inside the parentheses is raised to the power.

In the original problem, we have \( 2^{\frac{5}{2}} \), which was derived from \( \sqrt{2^5} \). Applying the property \( \sqrt{a^n} = a^{n/2} \), we simplified it as part of the expression, \( \frac{4\sqrt{2}}{\sqrt{3}} \). Recognizing and using these properties helps to transform complex numbers and makes calculations much easier.