Problem 43 - Algebra 2

Question

Write each expression in simplest form. Assume that all variables are positive. $$\left(-32 y^{15}\right)^{\frac{1}{3}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(-32 y^{15}\right)^{\frac{1}{3}}\) is \(-3y^5\).

Step 1: Break Down the Expression

The expression can be broken down to \(-32\) and \(y^{15}\), each separately raised to the power of \(1/3\). So the expression becomes \((-32)^{1/3} \cdot (y^{15})^{1/3}\).

Step 2: Take the Cube Root

Calculate the cube root of \(-32\), which is \(-3\) and the cube root of \(y^{15}\). Remember that when raising a power to a power, the powers should be multiplied, giving \(y^{15*1/3} = y^5\). So the expression simplifies to \(-3 \cdot y^5\).

Step 3: Put Together

The simplified form of the expression is \(-3y^5\).

Key Concepts

Cube Root

The cube root is an important concept when simplifying algebraic expressions, especially those involving exponents. In our exercise, the task is to find the cube root of a negative number, specifically \(-32\). Just as the square root of a number asks "what number, multiplied by itself twice, equals this value?"; the cube root asks "what number, multiplied by itself three times, equals this value?".

• The cube root of \(-32\) is \(-2\) because \(-2 \times -2 \times -2 = -8\).
• It may seem tricky since the result is negative, but that just indicates we need a negative number, because a negative multiplied by a negative yields a positive, yet another negative gets us back to negative.

Recognizing these patterns will help tremendously. The cube root allows us to simplify complex expressions by reducing the base numbers to their simplest form before handling the variables involved.

Exponents

Exponents are a way to express repeated multiplication of the same number or variable. For example, \(y^{15}\) means \(y\) multiplied by itself 15 times. In the context of our problem, we also use a fractional exponent, \(^{1/3}\). This fractional exponent signifies a root; specifically, that a cube root should be taken.

• When you see \(y^{15}\), and you apply an exponent of \(1/3\), you multiply the exponents: \(15\times \frac{1}{3} = 5\).
• This simplification turns \(y^{15}\) into \(y^5\). It's crucial to understand that fractional exponents are all about roots.

Understanding this concept allows us to tackle not only powers, but roots—whether square, cube, or higher. This is invaluable in breaking down complex expressions into manageable parts.

Simplification

Simplification is the art of transforming expressions into their most streamlined form. This makes them easier to understand, solve, or manipulate in further calculations.

• Our expression starts as \((-32y^{15})^{1/3}\).
• By breaking it down, we address each part separately: the constant \(-32\) and the variable expression \(y^{15}\).
• By using the cube root, we reduce \(-32\) to \(-2\), and transform \(y^{15}\) to \(y^5\).

Putting both results together leads to the simplest form: \(-2y^5\). Simplifying complex expressions often involves these steps—isolating different parts, performing necessary ruse of exponents, then combining them back for the final answer.