Problem 40 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{3} y^{5/7} \).

Step 1: Understanding Negative Exponents

Recall that a negative exponent indicates the reciprocal of the base. Here, we have a term with a negative exponent: \( y^{-5/7} \). This means \( y^{-5/7} \) is equivalent to \( \frac{1}{y^{5/7}} \).

Step 2: Rewriting the Expression

Using the understanding from Step 1, rewrite the expression by taking the reciprocal of \( y^{-5/7} \). This gives us: \[ \frac{2}{3} \times y^{5/7} \].

Step 3: Simplifying the Expression

You have rewritten the expression as \( \frac{2}{3} \times y^{5/7} \). Since there are no like terms or further simplification possible, this is the final simplified expression.

Key Concepts

Reciprocal

In mathematics, the term "reciprocal" simply means flipping the numerator and denominator of a fraction. It's like turning the fraction upside down. Now, when we deal with exponents, if the exponent is negative, the reciprocal of the base is used. For example, when we see something like \( y^{-1} \), it means we take the reciprocal of \( y \), ending up with \( \frac{1}{y} \). To help students apply this, let's look at our example: \( y^{-5/7} \). Because of the negative exponent, this can be rewritten using its reciprocal, which would be \( \frac{1}{y^{5/7}} \). So anytime you see a negative exponent, remember, you're really just flipping to find its reciprocal.

Simplification

Simplification in algebra is all about making expressions easier to work with. This often involves rewriting expressions in a cleaner and more straightforward way. With expressions that have negative exponents, like in our exercise, simplification often means converting these negative exponents into positive ones by using reciprocals.Let's go through the simplification step by step:

• Originally, we had \( \frac{2}{3 y^{-5/7}} \).
• By understanding negative exponents, we rewrite \( y^{-5/7} \) as \( \frac{1}{y^{5/7}} \).This changes our expression to \( \frac{2}{3} \times y^{5/7} \).

Now, in this expression, there are no further simplifications possible, so \( \frac{2}{3} \times y^{5/7} \) is considered simplified.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like addition or multiplication. Understanding these expressions is crucial for solving equations. Variables, represented by letters such as \( y \), can take on different values. Exponents tell us how many times to use the variable in a multiplication.In the context of our example:

• The variable is \( y \) and initially it has a negative exponent \( -5/7 \).
• We've learned to rewrite this using the reciprocal to turn the exponent positive, resulting in \( y^{5/7} \).

By manipulating algebraic expressions in this way, it becomes easier to solve equations and understand the mathematical relationships between different variables.