Problem 58 - Algebra A Combined Function

Question

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{8 a^{6} b^{9}} $$

Step-by-Step Solution

Answer

\( 2 a^2 b^3 \)

Step 1: Identify the Cube Root Expression

The problem asks you to simplify the cube root expression \( \sqrt[3]{8 a^6 b^9} \). This means we need to express the entire expression inside the cube root in a form that can be easily simplified.

Step 2: Simplify the Numerical Part

Start by taking the cube root of the numerical part of the expression. Here, the cube root of 8 is 2, because \( 2^3 = 8 \).

Step 3: Simplify the Variable Parts

For the variables, use the property \( \sqrt[3]{x^n} = x^{n/3} \). This way, \( \sqrt[3]{a^6} = a^{6/3} = a^2 \) and \( \sqrt[3]{b^9} = b^{9/3} = b^3 \).

Step 4: Combine the Results

Putting together the results from the previous steps, the expression \( \sqrt[3]{8 a^6 b^9} \) simplifies to \( 2 a^2 b^3 \).

Key Concepts

Cube Root Property

When we talk about cube roots, we mean finding a number (or expression) that, when multiplied by itself twice more, gives the original number. In algebraic terms, the cube root of a number \(x\) is written as \(\sqrt[3]{x}\). To simplify cube roots, we often use the property: \(\sqrt[3]{x^n} = x^{n/3}\). This property is helpful because it allows us to transform complex expressions into simpler terms by dealing with their exponents more conveniently.
In the original problem \(\sqrt[3]{8 a^6 b^9}\), each part of the expression can be separated. This means identifying the parts that can be simplified separately under the cube root, making it easier to deal with each component individually. The cube root property helps us break down each variable into a fraction of its original exponent so we can simplify easily.

• Identify and separate the different components under the cube root.
• Apply the cube root property to simplify each component's exponent.
• Combine these simpler components to achieve the final simplified expression.

Exponent Rules

Understanding exponent rules is like having a secret code to unlock the essence of algebraic simplification. Exponents tell us how many times a number, or variable, is multiplied by itself. When simplifying expressions involving roots and exponents, it is crucial to recall a few key rules.

• Addition and subtraction of exponents occur when multiplying or dividing like bases: for multiplication \(x^a \times x^b = x^{a+b}\) and for division \(x^a \div x^b = x^{a-b}\).
• Powers of powers are handled by multiplying exponents: \((x^a)^b = x^{a \, b}\).

In the context of the given problem, raise each term to the power of \(\frac{1}{3}\) using the cube root property. For example, when simplifying \(a^6\) within the cube root, it becomes \(a^{6/3} = a^2\). Understanding and applying these rules allow us to confidently simplify complex expressions by methodically breaking down each part of the operation.

Algebraic Simplification

Algebraic simplification is the process that helps us reduce complex equations or expressions to their simplest form. This process can involve factoring, combining like terms, or applying specific math properties, such as the cube root property or exponent rules. The end goal is to transform an unwieldy expression into a form that is easier to understand or use.
In the example provided, the expression \(\sqrt[3]{8 a^6 b^9}\) was simplified step-by-step:

• First, the numerical component \(8\) is simplified to \(2\) using the cube root, because \(2^3 = 8\).
• Next, the variable sub-expressions \(a^6\) and \(b^9\) were simplified using the exponent rules, resulting in \(a^2\) and \(b^3\) respectively.
• Finally, these results are combined to give the simplified expression \(2a^2b^3\).

Using these techniques not only simplifies your calculations but also makes the expressions more manageable for further mathematical operations.