Questions: Simplify. 3 y^-4 * 3 y^6 w^-5 u^-8 u * 9 w^5 Use only positive exponents in your answer.

Transcript text: Incorrect Your answer is incorrect. Simplify. \[ 3 y^{-4} \cdot 3 y^{6} w^{-5} u^{-8} u \cdot 9 w^{5} \] Use only positive exponents in your answer.

Solution

Solution Steps

To simplify the given expression, we need to apply the laws of exponents. First, combine the coefficients by multiplying them together. Then, for each variable, add the exponents of like bases. Finally, ensure all exponents are positive by using the property \(a^{-n} = \frac{1}{a^n}\).

Step 1: Combine Coefficients

First, we multiply the numerical coefficients: \(3 \times 3 \times 9 = 81\).

Step 2: Apply the Laws of Exponents

For each variable, we add the exponents of like bases:

For \(y\), the exponents are \(-4\) and \(6\). Thus, \(y^{-4} \cdot y^6 = y^{-4+6} = y^2\).
For \(w\), the exponents are \(-5\) and \(5\). Thus, \(w^{-5} \cdot w^5 = w^{-5+5} = w^0 = 1\).
For \(u\), the exponents are \(-8\) and \(1\). Thus, \(u^{-8} \cdot u = u^{-8+1} = u^{-7}\).

Step 3: Express with Positive Exponents

The expression \(u^{-7}\) can be rewritten with a positive exponent as \(\frac{1}{u^7}\).