Problem 75 - 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 expression simplifies to \(12a^{1/3}\).

Step 1: Simplify Cube Roots

First, identify the cube root components in each term. Each term has the form \(2 \times \sqrt[3]{x}\). Start with \(\sqrt[3]{64a}\). Since \(64 = 4^3\) and \(a = a^1\), we have \(\sqrt[3]{64a} = 4a^{1/3}\).

Step 2: Simplify the Second Cube Root

Next, look at \(\sqrt[3]{8a}\). Since \(8 = 2^3\), we have \(\sqrt[3]{8a} = 2a^{1/3}\).

Step 3: Factor the Expression

Combine the terms: \(2 \times \sqrt[3]{64a} + 2 \times \sqrt[3]{8a} = 2(4a^{1/3}) + 2(2a^{1/3})\). Factor out the common term \(2a^{1/3}\).

Step 4: Combine Like Terms

By factoring, the expression becomes \(2a^{1/3}(4 + 2)\). Calculate the sum inside the parenthesis: \(4 + 2 = 6\).

Step 5: Simplify the Expression

The expression simplifies to \(2a^{1/3} \times 6 = 12a^{1/3}\).

Key Concepts

Cube Roots

Understanding cube roots is crucial when dealing with algebraic expressions involving cubes. To find the cube root of a number, you are essentially looking for a value that, when multiplied by itself three times, gives the original number.
For instance, the cube root of 64 is 4, because multiplying 4 by itself three times (4 x 4 x 4) equals 64. This is often written mathematically as \(\sqrt[3]{64} = 4\).
Cube roots can also apply to variables in algebraic expressions. In the expression \(\sqrt[3]{64a}\), you can separate the number 64 and the variable \(a\), calculating their cube roots separately. If the variable is expressed as a power, like \(a^1\), you would keep \(a\) inside the root, giving \(\sqrt[3]{64}\times a^{1/3}\).
Knowing how to simplify cube roots not only helps in reducing the expression but also sets the stage for more complex manipulation and problem-solving in algebra.

Factoring Expressions

Factoring expressions involves breaking down a complex expression into simpler parts that can be multiplied together to give the original expression. This technique is particularly useful in simplifying algebraic fractions or solving equations.
In our exercise, once the cube roots are simplified, the expression \(2 \times \sqrt[3]{64a} + 2 \times \sqrt[3]{8a}\) splits into \(2(4a^{1/3}) + 2(2a^{1/3})\).
Here, both terms have a common factor, \(2a^{1/3}\). By factoring out \(2a^{1/3}\), the expression becomes \(2a^{1/3}(4 + 2)\). This step is important because it simplifies further calculations and enables combining like terms.
Factoring expressions often whether out numbers, variables, or both, helps in organizing the terms in a manageable way and sets up the expression for any needed simplification or solving.

Combining Like Terms

Combining like terms is a simplification technique used to make equations more manageable. It involves summing all terms that contain the same variable factors and it often requires prior steps, like factoring, to clearly see which terms can be combined.
In the exercise, after factoring out common terms, you're left with \(2a^{1/3}(4 + 2)\). To simplify this, perform the arithmetic inside the parentheses to get \(4 + 2 = 6\). This final step results in the simplified expression \(12a^{1/3}\).
When combining like terms, ensure that the terms share the same variables and powers. This approach reduces the complexity of expressions and leads to cleaner, simpler solutions. It's a key skill in algebra, allowing you to consolidate expressions and solve equations more efficiently.