Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$(\sqrt[3]{y^{2}})^{-5}$$

Step-by-Step Solution

Answer

\(\frac{1}{y^{10/3}}\)

Step 1: Understand the expression

We start with the expression \((\sqrt[3]{y^{2}})^{-5}\). The term \(\sqrt[3]{y^{2}}\) represents the cube root of \(y^2\).

Step 2: Convert to rational exponent

The expression \(\sqrt[3]{y^{2}}\) can be rewritten using rational exponents as \(y^{2/3}\). This follows from the fact that a cube root is equivalent to raising to the power of \(1/3\).

Step 3: Apply the negative exponent

Applying the negative exponent \(-5\) to \(y^{2/3}\), we have: \((y^{2/3})^{-5}\).

Step 4: Simplify using the power of a power rule

Use the exponent rule \((a^m)^n = a^{m \cdot n}\) to combine the exponents. We get \(y^{(2/3) \cdot (-5)} = y^{-10/3}\).

Step 5: Convert negative exponent to positive

Rewrite \(y^{-10/3}\) as a fraction: \(\frac{1}{y^{10/3}}\) to express the expression with a positive exponent.

Key Concepts

Rational Exponents

In mathematics, rational exponents provide a way to express roots and powers within a single notation. Instead of writing the cube root of something, like \( \sqrt[3]{y^2} \), you can use a fraction as the exponent: \( y^{2/3} \). Here’s why and how it works:

• The numerator (top part) of the rational exponent tells you the power. In \( y^{2/3} \), the 2 tells us that we are squaring \( y \).
• The denominator (bottom part) tells you the root. The 3 indicates a cube root.

Using rational exponents simplifies calculations and unifies the operations of taking roots and raising powers. It also prepares you for more advanced math topics where these concepts are used extensively.

Negative Exponents

Negative exponents might seem tricky at first, but they have a simple rule: a negative exponent means "one over" the positive exponent. It's helpful because it offers a quick way to move terms with exponents to different parts of a fraction.
To make sense of \( y^{-10/3} \), remember:

• A negative exponent indicates the reciprocal. So, \( y^{-1} = \frac{1}{y} \).
• Therefore, \( y^{-10/3} = \frac{1}{y^{10/3}} \).

This process of turning negative exponents into positive ones helps with simplification. It can also make expressions easier to evaluate, especially when combining different terms.

Exponent Rules

Exponent rules are guidelines that make it easier to handle expressions with powers. Knowing key rules streamlines simplification and manipulation of powers.
Here are some important ones:

• The Product Rule: \( a^m \times a^n = a^{m+n} \).
• The Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \).
• The Power Rule: \((a^m)^n = a^{m \cdot n} \).

In the step where we simplified \( (y^{2/3})^{-5} \) to \( y^{-10/3} \), we used the power rule by multiplying the exponents. These rules are fundamental tools when working with exponents, allowing for quick and accurate transformations of expressions.