Problem 24 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{288 x^{4}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(12x^2\).

Step 1: Factor Under the Radical

First, we need to factor the number 288 completely to help us simplify the radical. The factorization of 288 is: \[ 288 = 2^5 \times 3^2. \] For the variable part \(x^4\), it is already expressed as powers of a variable.

Step 2: Simplify the Radical Expression

Using the property that \(\sqrt{a^2} = a\) (assuming \(a\) is non-negative), we can take even powers out of the radical:\[ \sqrt{288 x^4} = \sqrt{2^5 \times 3^2 \times x^4}. \] We can take out the square root of \(3^2\) which is \(3\) and \(x^4\) as \(x^2\) since \(\sqrt{x^4} = x^2\). This simplifies to:\[ 3x^2 \times \sqrt{2^4}. \]

Step 3: Simplify Further

Continue simplifying by breaking down \(\sqrt{2^4}\).\[ \sqrt{2^4} = 2^2 = 4. \] This means we can take \(2^2\) out as 2 from \(\sqrt{2^4}\) term:\[ \sqrt{2^4} = 2^2 = 4 \].

Step 4: Combine All Parts

Now multiply the numbers from outside the radical \[ 3 \times 2 \times x^2 = 6x^2 \] Thus, the expression \(\sqrt{2^5 \times 3^2 \times x^4}\) simplifies to:\[ 12x^2 \].

Key Concepts

Factorization

When dealing with radicals, one of the most fundamental steps is factorization. This involves breaking down a number into its prime factors, which are numbers that have no other divisors besides 1 and themselves. For the given number 288, the prime factorization is:

• 288 divided by 2 gives 144
• 144 divided by 2 gives 72
• 72 divided by 2 gives 36
• 36 divided by 2 gives 18
• 18 divided by 2 gives 9
• 9 divided by 3 gives 3
• 3 divided by 3 gives 1

So, 288 can be expressed as \[ 2^5 \times 3^2 \].This prime factor breakdown is crucial, as it allows us to more easily identify which parts of the expression under the radical can be simplified or moved out.

Properties of Exponents

Exponents are also pivotal in simplifying radicals, especially when dealing with variables. The basic rule to remember here is that taking the square root of a number is equivalent to raising it to the power of one half. For example, \( \sqrt{x^4} \) becomes \( x^{4/2} = x^2 \).When numbers or variables are raised to an even power, their square roots can be fully simplified out of the radical. Consider the factors of 288, specifically \( 3^2 \). The square root of \( 3^2 \) is simply \( 3 \).These rules apply when the expression's exponent is a multiple of two:

• \( x^4 \) becomes \( x^2 \) when square-rooted
• \( a^{2n} \) simplifies to \( a^n \)
• \( b^{2n+1} \) would have \( b^n \) taken out, leaving \( b \) inside the root

By understanding these rules, the entirety of the expression \( \sqrt{288 x^4} \) can be simplified step by step efficiently.

Square Roots

The square root, notated as \( \sqrt{} \), is one of the most common radical expressions. Its main property is that it "undoes" the effect of squaring a number. Suppose you have a perfect square such as 16; the square root is 4 because \( 4^2 = 16 \).Things become more interesting when the number is not a perfect square. In our exercise, after factorization, we break down the expression inside the square root into manageable parts:

• \( \sqrt{288 x^4} = \sqrt{ 2^5 \times 3^2 \times x^4 } \)

The parts which can be extracted directly, such as \( 3^2 \) and \( x^4 \), simplify to \( 3 \) and \( x^2 \) respectively. This gives us part of the result outside of the radical.The remaining factor, \( 2^4 \), still under the root, can further be simplified since \( \sqrt{2^4} = 2^2 \) equals 4. Consolidating these results helps fully resolve the original radical expression, showing the power and utility of carefully applying square root principles.