Problem 80 - Algebra A Combined Function

Question

Assume that all variables represent positive real numbers. $$ -\sqrt[3]{\frac{64 a^{3}}{b^{9}}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{4a}{b^3}\).

Step 1: Simplify the Expression Inside the Cube Root

The given expression is \(-\sqrt[3]{\frac{64 a^3}{b^9}}\). First, we need to simplify the expression inside the cube root. Notice that \(64 = 4^3\), so the expression becomes \(-\sqrt[3]{\frac{(4a)^3}{b^9}}\).

Step 2: Apply the Cube Root Property

Using the property \(\sqrt[3]{\frac{x^3}{y^3}} = \frac{x}{y}\), we can apply it to \(\frac{(4a)^3}{b^9}\) which gives: \(\frac{4a}{b^3}\).

Step 3: Include the Negative Sign

Finally, since the original expression has a negative sign outside of the cube root, we attach the negative sign to the simplified expression: \(-\frac{4a}{b^3}\).

Key Concepts

Cube Roots

A cube root finds a number which, when multiplied by itself three times, results in the original number. For example, the cube root of 8 is 2 because

• \(2 \times 2 \times 2 = 8\)

In this exercise, we worked with the cube root of a fraction:

• \(-\sqrt[3]{\frac{64 a^{3}}{b^{9}}}\)

The cube root simplifies and allows us to "divide" the powers by 3 inside the fraction.
This means that the power of \(a\) and \(b\) will decrease, simplifying the expression to easier terms.
Recognizing cube numbers like 64, as \(4^3\), helps to quickly break down the equation. This mathematical intuition is useful when dealing with roots of variables and numbers.

Simplification

Simplification in algebra involves reducing expressions into their simplest form. Think of it as cleaning up cluttered math. In this particular solution, the expression inside the cube root \(-\sqrt[3]{\frac{64 a^{3}}{b^{9}}}\) was simplified.
The first step was identifying \(64\) as \(4^3\), allowing the form \(\frac{(4a)^3}{b^9}\).
This clever manipulation paves the way for applying cube root rules. Then, using the property \(\sqrt[3]{\frac{x^3}{y^3}} = \frac{x}{y}\), the expression turns into a much simpler form.This procedure is all about recognizing patterns and mathematical properties:

• Identify powers that can be rooted.
• Match pairs or triplets of factors for roots.
• Use identities like \(a^3\) or \(b^9\) to simplify.

Consider simplification as finding the easiest way to express a complex idea in algebra.

Positive Real Numbers

In this exercise, all variables are assumed to represent positive real numbers.
This assumption is crucial for simplifying expressions involving roots.
Positive real numbers are all the non-negative numbers on a number line, excluding zero, that we typically deal with in everyday mathematics. Why is this assumption important?

• With positive values, finding real roots, like cube roots, remains straightforward, without involving complex or imaginary numbers.
• It ensures that variables under roots are valid, rendering clear simplifications.

Roots and powers behave predictably under positive numbers. For instance, \(a^3\) and \(b^9\) have roots because positive bases lead to positive results.
This assumption directly impacts the problem-solving strategy, ensuring the solution remains in the realm of real numbers, making calculations manageable and relatable.