Math Problem Statement

Simplify the expression \(2w^{-5} \cdot 7u^{-4} \cdot y^6 \cdot w^9 \cdot 2u^{-4}\) using only positive exponents.

Solution

Let’s simplify the given expression step by step.

The expression is:

2wy57u4y6w92u4.2w^{y^{-5}} \cdot 7u^{-4} \cdot y^6 \cdot w^9 \cdot 2u^{-4}.

Step 1: Combine like terms (grouping same bases)

  1. For ww:

    • wy5w9=wy5+9w^{y^{-5}} \cdot w^9 = w^{y^{-5} + 9}.
  2. For uu: (foul); It seems like I started off-track when combining terms due to notation confusion. Let’s simplify correctly:

The expression given is:

2wy57u4y6w92u4.2w^{y^{-5}} \cdot 7u^{-4} \cdot y^6 \cdot w^9 \cdot 2u^{-4}.

Let me interpret and rewrite it step by step.

Step 1: Simplify constants

Combine the numerical coefficients: 272=28.2 \cdot 7 \cdot 2 = 28.

So, the updated expression is: 28w5w9y6u4u4.28 \cdot w^{-5} \cdot w^9 \cdot y^6 \cdot u^{-4} \cdot u^{-4}.

Step 2: Combine powers for like bases

  • For ww: Add the exponents 5-5 and 99: w5+9=w4.w^{-5 + 9} = w^4.

  • For uu: Add the exponents 4-4 and 4-4: u44=u8.u^{-4 - 4} = u^{-8}.

  • yy remains as y6y^6.

Step 3: Use only positive exponents

  • u8u^{-8} becomes 1u8\frac{1}{u^8}.

So, the fully simplified expression is: 28w4y6u8.\frac{28w^4y^6}{u^8}.

Let me know if you have questions or would like further clarification!

Related Questions:

  1. How can I verify if my simplified expression is correct?
  2. What are the rules for adding and subtracting exponents when simplifying?
  3. How do negative exponents translate into fractions?
  4. Are there any shortcuts for simplifying expressions with multiple variables?
  5. Can we visualize how exponents combine using expanded forms?

Tip: Always combine like terms by ensuring all variables with the same base are grouped together before simplifying.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponent Rules
Algebraic Simplification

Formulas

Product of powers: \(a^m \cdot a^n = a^{m+n}\)
Negative exponents: \(a^{-m} = \frac{1}{a^m}\)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 8-10