Problem 26 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The expression is rewritten as \(\frac{1}{y^{\frac{10}{3}}}\).

Step 1: Rewrite the Radicals Using Exponents

First, realize that a cube root can be rewritten using exponents: \[\sqrt[3]{y^2} = y^{\frac{2}{3}}\]

Step 2: Apply the Power of a Power Rule

The expression given is \((\sqrt[3]{y^2})^{-5}\). Substitute the rewritten form of the radicals from Step 1:\[(y^{\frac{2}{3}})^{-5}\]When you raise a power to another power, you multiply the exponents.

Step 3: Multiply the Exponents

Following the power of a power rule, multiply the exponents:\[\left(y^{\frac{2}{3}}\right)^{-5} = y^{\left(\frac{2}{3}\right) \times (-5)} = y^{-\frac{10}{3}}\]

Step 4: Simplify Using Negative Exponents

Rewrite the expression with a positive exponent by taking the reciprocal:\[y^{-\frac{10}{3}} = \frac{1}{y^{\frac{10}{3}}}\]

Key Concepts

Cube Roots

Imagine you have a number, and you want to find another number that, when multiplied by itself twice, gives you the original number. That's what cube roots are all about! The cube root of a number is what you'd call the reverse of cubing a number.
Here's how it works:

• Cubing a number means multiplying it by itself three times. For instance, if you multiply 2 by itself three times, 2 × 2 × 2 = 8, then 2 is the cube root of 8.
• In terms of symbols, the cube root of a number 'x' is written as \(\sqrt[3]{x}\).
• This can also be expressed using rational exponents: \(x^{\frac{1}{3}}\) is the same as the cube root of \(x\).

Cube roots can help simplify algebraic expressions and solve equations, as we often rewrite them using exponents for easier computation.

Negative Exponents

Negative exponents might seem puzzling at first, but they're not too tricky once you get the hang of it! A negative exponent indicates that the base is on the wrong side of a fraction and needs to be flipped.
Let's break it down:

• A negative exponent, like \(x^{-n}\), means you take the reciprocal of \(x\) and then raise it to the positive exponent: \(x^{-n} = \frac{1}{x^n}\).
• Essentially, you are "turning the number upside down." For example, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).
• This concept allows you to work with equations and expressions by turning multiplication into division, through which simplification can be achieved more seamlessly.

Mastering negative exponents is crucial in algebra, as it turns tough-looking expressions into something more manageable!

Power of a Power Rule

The power of a power rule is a handy tool for simplifying expressions that involve multiple exponents. It tells us how to handle a variable that is raised to one power and then raised to another.
Here's the drill:

• When you see something like \((x^m)^n\), apply the rule by multiplying the exponents: \(x^{m \times n}\).
• This rule comes in particularly handy when you're dealing with expressions such as \( (y^{\frac{2}{3}})^{-5} \), where you multiply \(\frac{2}{3} \times (-5) \) to get \(-\frac{10}{3}\).
• It simplifies the steps needed to tackle algebraic expressions that would otherwise be time-consuming to solve.

Having this in your toolkit means you can breeze through complex expressions that involve layers of powers, making problems easier and faster to solve.