Problem 40 - Algebra A Combined Function

Question

Write with positive exponents. Simplify if possible. $$ \frac{2}{3 y^{-5 / 7}} $$

Step-by-Step Solution

Answer

\( \frac{2y^{5/7}}{3} \) is the simplified expression using positive exponents.

Step 1: Rewrite with Positive Exponents

The given expression is \( \frac{2}{3y^{-5/7}} \). To rewrite it using positive exponents, we need to apply the property that \( a^{-n} = \frac{1}{a^n} \). Therefore, \( y^{-5/7} \) can be written as \( \frac{1}{y^{5/7}} \). This modifies the expression to \( \frac{2}{3 \cdot \frac{1}{y^{5/7}}} \).

Step 2: Simplify the Expression

Simplify the expression \( \frac{2}{3 \cdot \frac{1}{y^{5/7}}} \). This is equivalent to \( \frac{2}{3} \times \frac{y^{5/7}}{1} \). Multiplying these fractions, you get \( \frac{2 \cdot y^{5/7}}{3} \).

Step 3: Final Simplification

After the multiplication, the expression becomes \( \frac{2y^{5/7}}{3} \). There are no more terms to simplify, so this is the expression in its simplest form using positive exponents.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. It's important to become comfortable with these expressions, as they form the basis of algebra. In this exercise, we are dealing with an expression that involves a fraction and negative exponents.Here's a quick rundown of what's in an algebraic expression:

• **Variables**: These are symbols, like \( y \), that can represent numbers.
• **Constants**: These are fixed numbers, like \( 2 \) and \( 3 \) in our example.
• **Operations**: These include multiplication and division, as well as applying exponents.

When working with algebraic expressions, understanding each component's role is essential. Grasping how to manipulate them—such as converting negative exponents or simplifying fractions—helps in simplifying and solving mathematical problems.

Exponent Rules

Exponent rules play a crucial role in simplifying algebraic expressions that include powers. The rules help us manage and manipulate expressions efficiently. A key concept here is understanding negative exponents and how to convert them to positive.Here's what you need to know about the key exponent rule used in this problem:

• **Negative Exponent Rule**: When you see a negative exponent, like \( y^{-5/7} \), it can be rewritten. The rule states \( a^{-n} = \frac{1}{a^n} \). So, \( y^{-5/7} \) becomes \( \frac{1}{y^{5/7}} \).

By converting negative exponents to positive, we can simplify expressions more easily. This makes calculations more straightforward and opens up further simplification opportunities, like in our exercise.

Fraction Simplification

Fraction Simplification involves combining and reducing fractions to their simplest form. Let's break down how we used these principles in the problem we solved.Our original expression, \( \frac{2}{3 y^{-5/7}} \), required the transformation of a component within the fraction. After rewriting \( 3y^{-5/7} \) as \( 3 \cdot \frac{1}{y^{5/7}} \), we can tackle the fraction simplification.Here's a step-by-step of how it simplifies:

• **Multiply Across**: When you have \( \frac{2}{3} \times \frac{y^{5/7}}{1} \), you multiply directly to get \( \frac{2 \cdot y^{5/7}}{3} \).

There are no further terms to reduce, meaning the fraction is now in its simplest form. Mastering fraction simplification is essential to dealing with complex algebraic expressions, as it makes working with them much less intimidating.