Problem 75 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ 2 \sqrt[3]{64 a}+2 \sqrt[3]{8 a} $$

Step-by-Step Solution

Answer

The simplified expression is \(12\sqrt[3]{a}\).

Step 1: Separate the Terms

Identify and separate the two cube root terms in the expression:1. \(2 \sqrt[3]{64a}\)2. \(2 \sqrt[3]{8a}\).

Step 2: Simplify Cubes

Recognize that both 64 and 8 are perfect cubes (\(64 = 4^3\) and \(8 = 2^3\)).This lets us simplify each part by calculating the cube roots:- \(\sqrt[3]{64a} = \sqrt[3]{4^3a} = 4\sqrt[3]{a}\)- \(\sqrt[3]{8a} = \sqrt[3]{2^3a} = 2\sqrt[3]{a}\).

Step 3: Plug Back the Simplified Values

Insert the new values back into the original expression:\(\begin{align*}2 \sqrt[3]{64a} + 2 \sqrt[3]{8a} &= 2(4\sqrt[3]{a}) + 2(2\sqrt[3]{a}) \&= 8\sqrt[3]{a} + 4\sqrt[3]{a}\end{align*}\).

Step 4: Combine Like Terms

Since both terms have \(\sqrt[3]{a}\), they can be combined:\(8\sqrt[3]{a} + 4\sqrt[3]{a} = (8 + 4)\sqrt[3]{a} = 12\sqrt[3]{a}\).

Key Concepts

Understanding Cube Roots

A cube root is a number that, when multiplied by itself twice, results in the original number under the radical. For example, the cube root of 8, written as \( \sqrt[3]{8} \), is 2 because \( 2 \times 2 \times 2 = 8 \). Cube roots help us simplify expressions that contain terms raised to the power of three.
Most importantly, to find a cube root, you need to identify number factors that repeat three times. For instance:

• For \(64\), since \(4^3 = 64\),
  the cube root is 4: \(\sqrt[3]{64} = 4\).
• Similarly, \(2^3 = 8\) implies that \(\sqrt[3]{8} = 2\).

Cube roots are useful in algebraic simplification as they allow breaking down complex expressions into manageable parts. Recognizing them simplifies computations and combining terms. Always look for numbers that are raised to the power of three to identify cube roots in an expression.

Exploring Perfect Cubes

Perfect cubes are numbers that result from multiplying an integer by itself two additional times, forming a cubic volume. These numbers have integers as their cube roots. Common examples include:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)

In algebra, recognizing perfect cubes is vital for simplifying expressions.
This is because we can more easily take cube roots from them. When you see a number in an expression, checking if it's a perfect cube helps simplify your work by letting you replace it with its cube root value.
For example, in the expression \(2 \sqrt[3]{64a}\), we identify \(64\) as a perfect cube, making our calculations more straightforward: \(\sqrt[3]{64} = 4\). Understanding perfect cubes plays a crucial role in algebraic manipulation by seamlessly simplifying expressions.

Simplifying by Combining Like Terms

Combining like terms is a fundamental aspect of simplifying algebraic expressions. Terms are considered "like" if they have identical variable parts. In this context, it means they must have the same cube root expressions, such as \(\sqrt[3]{a}\).Let's examine the expression:
\(8\sqrt[3]{a} + 4\sqrt[3]{a}\).
Both terms share \(\sqrt[3]{a}\) as a factor, making them like terms. We can add their coefficients (8 and 4) directly:

• \(8 + 4 = 12\).

Therefore, the expression simplifies to \(12\sqrt[3]{a}\). The goal of combining like terms is to make expressions more concise and easier to work with. It's a straightforward way to tidy up algebraic expressions by reducing them to a simpler form. Remember, this step only works when the variable components, specifically the cube roots, are identical. Ensuring students understand this can greatly improve their algebraic simplification skills.