Problem 44 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[5]{32 z^{12}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2z^{\frac{12}{5}}\).

Step 1: Rewrite the Radical as an Exponent

The expression \( \sqrt[5]{32 z^{12}} \) can be rewritten using exponents. The fifth root of a term is equivalent to raising that term to the power of \( \frac{1}{5} \). Hence, \( \sqrt[5]{32 z^{12}} = (32 z^{12})^{\frac{1}{5}} \).

Step 2: Apply the Power to Each Factor

Using the property of exponents that states \((ab)^m = a^m b^m\), apply the exponent to each factor: \((32 z^{12})^{\frac{1}{5}} = 32^{\frac{1}{5}} z^{12 \cdot \frac{1}{5}} \).

Step 3: Simplify Each Term

Now simplify each term individually. Firstly, \(32^{\frac{1}{5}} = 2\) because 32 is \(2^5\). For the second term, \(z^{12 \cdot \frac{1}{5}} = z^{\frac{12}{5}}\). This gives us the simplified expression: \(2z^{\frac{12}{5}}\).

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. For instance, if you have the exponent notation \(a^n\), the base \(a\) is multiplied by itself \(n\) times. So, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\). When dealing with problems like the fifth root or any roots, they can be expressed as fractions. For our exercise, the fifth root is expressed as \(\frac{1}{5}\) in the exponent form.
This makes it easier to manipulate radical expressions and simplify them using exponent rules. This shows that different ways of presenting numbers, such as exponential or radical, are deeply connected. Understanding this connection helps in converting between formats, making calculations easier. Always remember that the numerator of a fractional exponent gives the power and the denominator gives the root.

Simplifying Radicals Made Simple

Simplifying radicals involves converting a radical expression into an exponential form and then reducing it using the properties of exponents. Consider the radical \(\sqrt[5]{32 z^{12}}\).
First, rewrite it as \((32 z^{12})^{\frac{1}{5}}\). This translates the radical into a more standard form of exponential notation which is easier to work with mathematically.

• We start with \((32 z^{12})^{\frac{1}{5}}\).
• Apply the exponent to each element: \(32^{\frac{1}{5}} z^{12 \cdot \frac{1}{5}}\).
• Finding \(32^{\frac{1}{5}} = 2\), since 32 is the same as \(2^5\).
• The variable term becomes: \(z^{\frac{12}{5}}\).

Thus, the expression simplifies overall to \(2z^{\frac{12}{5}}\). By breaking down radicals into exponents, simplifying becomes systematic and efficient.

Properties of Exponents Explained

Understanding the properties of exponents is key to simplifying expressions efficiently. These properties allow us to manipulate exponential expressions to make calculations simpler.
Key properties include:

• Quotient Rule: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)

In our exercise, the property \((ab)^m = a^m b^m\) was crucial. It allowed us to distribute the exponent to both 32 and \(z^{12}\), breaking them down into simpler components that can be solved independently.
This method showcases the power of understanding properties of exponents, making what initially might seem complex a much more straightforward task. Always keep these rules in mind when working with exponents to boost your mathematical problem-solving skills.