Problem 30 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{1000 a}{b^{9}}} $$

Step-by-Step Solution

Answer

\(-\frac{10a^{1/3}}{b^3}\)

Step 1: Rewrite the Expression Using Exponents

Rewrite the original expression \(-\sqrt[3]{\frac{1000a}{b^9}}\)in terms of exponents: \(-\left(\frac{1000a}{b^9}\right)^{1/3}\).

Step 2: Simplify Inside the Radical

First, simplify the expression inside the cube root: \(\frac{1000a}{b^9}\) can be split into \(1000^{1/3}\times a^{1/3}\times b^{-9/3}\).

Step 3: Evaluate the Cube Root of Numerical Values and Simplify

Calculate \(1000^{1/3}\): The cube root of 1000 is 10, so \(1000^{1/3} = 10\).

Step 4: Apply Exponents to Each Part in the Expression

Express each term with exponents: cube root of the entire expression becomes, \(-10 \cdot a^{1/3} \cdot b^{-3}\).

Step 5: Combine the Exponents

Combine the components together: Thus, the expression simplifies to \(-\frac{10a^{1/3}}{b^3}\).

Key Concepts

Simplifying Expressions

Let's talk about simplifying expressions. It's like tidying up a messy room full of mathematical terms. The goal is to make the expression as simple and neat as possible by following specific rules.
One common rule is the Quotient Rule, especially when you have terms in a fraction. Here, the key step is to find a common base and simplify powers using exponent rules. You start by consolidating like terms and reducing everything to its simplest form.

• Identify and separate numerical and variable parts.
• Check for opportunities to apply mathematical rules, such as exponent rules.
• Simplify step-by-step, keeping track of what changes at each step.

The more you simplify an expression, the easier it becomes to understand and work with. The final expression should be easy to read and manageable for further mathematical operations.

Exponents

Understanding exponents is essential in math, as they represent repeated multiplication of a number by itself. When simplifying expressions with exponents, knowing the basic rules can be incredibly helpful.
The example involves taking cube roots, which is related to exponents. The cube root can be expressed as a power of one-third, \(^{1/3}\). The original exercise expression is rewritten using exponents as \(-\left(\frac{1000a}{b^9}\right)^{1/3}\).

• To simplify, separate each term in the fraction and apply the power \(^{1/3}\) to them individually.
• For numerical values, like 1000, calculate using \(1000^{1/3} = 10\), because \(10^3 = 1000\).
• For variables, apply the fraction to the exponent: \(a^{1}\cdot a^{1/3}\) and \((b^{-9/3})\).

These operations follow exponent rules like power of a product, which allows us to deal with each component separately and then combine them into a simplified form, \(-10 \, a^{1/3} b^{-3}\).

Cube Roots

Cube roots are like the inverse of cubing a number or expression. They undo a cube, telling us what number, when multiplied by itself three times, gives us the original number. This is crucial when simplifying expressions involving cube roots, such as in our example.
To find the cube root of a number, identify the base number that, multiplied by itself three times, equals that number. For instance, in the expression \(\sqrt[3]{1000}\) becomes \(10\) because \(10^3 = 1000\).

• Cube roots are helpful for simplifying expressions where entire terms are cubed.
• They are written as a number or expression within a radical symbol with a small three signifying the cube root.
• Cube roots can be expressed using fractional exponents, offering more versatility in solving complex expressions.

In our case, dealing with cube roots as fractional exponents allows us to apply the power \((^{1/3})\) across each term; simplifying it in a step-by-step manner results in a cleaner, more comprehensible expression.