Problem 35 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[3]{16(-2)^{4}(2)^{8}}$$

Step-by-Step Solution

Answer

The simplified expression is \(32 \sqrt[3]{2}\).

Step 1: Simplify the expression inside the radical

Start by simplifying the expression inside the cube root: \(16(-2)^4(2)^8\). 1. Calculate \((-2)^4\): \((-2)^4 = 16\). 2. Calculate \((2)^8\): \((2)^8 = 256\). Now the expression becomes \(16 \times 16 \times 256\).

Step 2: Simplify the product

Now multiply the simplified numbers inside the cube root:\(16 \times 16 = 256\).Next, multiply \(256 \times 256\):\(256 \times 256 = 65536\).So, the expression now is \(\sqrt[3]{65536}\).

Step 3: Find the cube root

Now determine the cube root of 65536. Recall that:\(65536 = 2^{16}\).The cube root of \(2^{16}\) is calculated as follows:\(\sqrt[3]{2^{16}} = 2^{\frac{16}{3}} = 2^{5 + \frac{1}{3}}\).This simplifies to \(32 \times \sqrt[3]{2}\).

Key Concepts

Cube Roots

Understanding cube roots is essential when simplifying radical expressions like the one in our example, \(\sqrt[3]{16(-2)^4(2)^8}\). A cube root is simply a number that, when multiplied by itself three times, results in the given number. To visualize:

• The cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
• The cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\).

When simplifying a cube root, you look for factors that can be expressed as perfect cubes. In the problem, once simplified, the expression inside became \(65536\), which can be written as \(2^{16}\). Hence, finding the cube root involves figuring out which number cubed equals \(2^{16}\). Breaking it down, \(2^{16}\) means finding the cube root \(\sqrt[3]{2^{16}}\), which ultimately simplifies using the exponent rules. This results in \(32 \times \sqrt[3]{2}\), where \(32\) is a number that you can multiply by itself three times to achieve a part of the original number.

Exponents

Exponents are the shorthand used for expressing the repeated multiplication of a number by itself. They are crucial in simplifying radical expressions. Consider the original expression \(16(-2)^4(2)^8\). Here, exponents show how many times each base is used as a factor. Let's break this down further:

• \((-2)^4 = 16\) indicates \((-2)\) is multiplied by itself 4 times.
• \((2)^8=256\) means \(2\) is used as a factor 8 times repeated.

By knowing exponents, you can easily convert large products into a more manageable form. For example, once all factors in the expression are multiplied, we achieve \(65536\), expressed as \(2^{16}\). Understanding exponents allows you to apply rules that break down expressions into simpler components, helping in finding cube roots or any radical manipulations required.

Product Property of Radicals

The product property of radicals is a vital tool when simplifying radical expressions involving multiplication. This property states that the radical of a product is equal to the product of the radicals of each factor. In simpler terms, if you have a product under a radical, you can split it into separate radicals for each factor.For example, if you have \(\sqrt{a \times b}\), it is equivalent to \(\sqrt{a} \times \sqrt{b}\). In the case of cube roots, it looks like:\[ \sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b} \]Applying this in our problem, after simplifying inside the cube root, we used the property to express the square number \(65536\) as \(2^{16}\), which helps us determine the radical. Knowing this property lets you break down complex expressions into smaller, more manageable parts. It's especially handy when individual factors can be simplified or have recognizable cube roots, streamlining the calculation process.