Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt[3]{128}+3 \sqrt[3]{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(7\sqrt[3]{2}\).

Step 1: Simplify the cube roots

Start by simplifying the cube root of 128. Notice that 128 can be expressed as a power of 2: \[ 128 = 2^7 \] Taking the cube root, we have: \[ \sqrt[3]{128} = \sqrt[3]{2^7} = 2^2 \times \sqrt[3]{2} = 4 \sqrt[3]{2} \]

Step 2: Combine like terms

After simplifying \( \sqrt[3]{128} \) as \( 4\sqrt[3]{2} \), we can combine it with the other term \( 3\sqrt[3]{2} \): \[ 4\sqrt[3]{2} + 3\sqrt[3]{2} = (4 + 3)\sqrt[3]{2} = 7 \sqrt[3]{2} \]

Step 3: Final expression

The simplified expression is now: \[ 7\sqrt[3]{2} \]. This is the final simplified form of the given expression.

Key Concepts

Simplifying Radicals

Radicals often appear in algebra when dealing with roots. Simplifying them makes equations easier to work with. The goal is to express roots, like square or cube roots, in their simplest form. Here's how you can simplify cube roots, which are an extension of square roots into the third dimension.
When simplifying a radical such as \[\sqrt[3]{128}\], the first step is to express the number under the radical as a power of a smaller number. You do this because cube roots simplify neatly when the number is a perfect cube or can be factored into a perfect cube and another number.

• First, factor the number completely. For 128, notice it can be expressed as \(2^7\).
• Since cube roots mean we're looking for something times itself three times, break \(2^7\) down into \(2^6 \times 2\), or \((2^3)^2 \times 2\).
• Then, apply the cube root: \(\sqrt[3]{2^7} = 2^2 \times \sqrt[3]{2} = 4 \sqrt[3]{2}\).

This simplification reduces the complexity of the radical, allowing for easier arithmetic when combining terms later on.

Cube Roots

Cube roots are a specific type of radical where you're looking for a number which, when multiplied by itself twice more, returns the original number. This is analogous to square roots, but for three dimensions, or cubes.
For instance, finding \(\sqrt[3]{x}\)aims to discover \(y\) such that \(y \times y \times y = x\). This is a bit more complex than square roots because there is another layer of multiplication involved.
To deal with cube roots effectively, it's crucial to understand:

• How to rewrite numbers as powers, especially focusing on expressions involving primes like 2 and 3.
• You can simplify any cube root by identifying parts of the number that are perfect cubes. For example, with \(\sqrt[3]{128}\), we look for the largest cube inside 128, which is \(64\).
• The result after simplification will usually be in the form of a whole number times a radical, which is much easier to manage in an expression.

Recognizing the structure and simplifying cube roots can lead to streamlined calculations and clearer results.

Combining Like Terms

Once you've simplified the radicals, it's time to combine terms that are alike. In algebra, "like terms" are terms that have the same variables raised to the same powers with the same radical part. Combining them makes an expression much simpler and easier to understand.
Let's take a look at the expression \(4\sqrt[3]{2} + 3\sqrt[3]{2}\).
Because these terms have the identical radical component \(\sqrt[3]{2}\), they can be added, just like adding \(4x\) and \(3x\) to get \(7x.\)

• Add the coefficients of the like terms, in this case, \(4\) and \(3\).
• The simplified expression becomes \(7\sqrt[3]{2}\), thanks to the same radical factor.

The concept of combining like terms not only applies to radicals but also to algebraic terms, and it's essential for solving and simplifying more complex expressions.