algebra 1
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product and quotient rules exit ticket
simplify. your answer should contain only positive exponents.
1) ( 3b^2 cdot 3b^2 cdot 2b )
2) ( y^2 cdot 2x^3 y^3 )
3) ( \frac{3x}{x^5} )
4) ( \frac{2yx^3}{3x^2 y^3} )

Question

Accepted Answer

### Problem 1: $ 3b^2 \cdot 3b^2 \cdot 2b $
# Explanation:
## Step 1: Multiply coefficients
Multiply the numerical coefficients: $ 3 \times 3 \times 2 = 18 $
## Step 2: Apply exponent rule for multiplication ($ a^m \cdot a^n = a^{m + n} $)
For the variable $ b $: $ b^2 \cdot b^2 \cdot b = b^{2 + 2 + 1} = b^5 $
## Step 3: Combine results
Multiply the coefficient and the variable term: $ 18 \times b^5 = 18b^5 $
# Answer: $ 18b^5 $

### Problem 2: $ y^2 \cdot 2x^3y^3 $
# Explanation:
## Step 1: Multiply coefficients
The coefficient of $ y^2 $ is 1, and the coefficient of $ 2x^3y^3 $ is 2. So, $ 1 \times 2 = 2 $
## Step 2: Apply exponent rule for multiplication on $ y $ terms
$ y^2 \cdot y^3 = y^{2 + 3} = y^5 $
## Step 3: The $ x $ term remains as $ x^3 $ (since there's only one $ x $ term)
## Step 4: Combine all terms
Multiply the coefficient, $ x $ term, and $ y $ term: $ 2 \times x^3 \times y^5 = 2x^3y^5 $
# Answer: $ 2x^3y^5 $

### Problem 3: $ \frac{3x}{x^5} $ (assuming the problem is $ \frac{3x}{x^5} $ as the image shows $ \frac{3x}{x^5} $)
# Explanation:
## Step 1: Apply exponent rule for division ($ \frac{a^m}{a^n} = a^{m - n} $)
For the variable $ x $: $ \frac{x}{x^5} = x^{1 - 5} = x^{-4} $
## Step 2: Rewrite with positive exponent ($ a^{-n} = \frac{1}{a^n} $), but we can also simplify the coefficient and the $ x $ term directly
Alternatively, we can think of $ \frac{3x}{x^5} = 3 \times \frac{x}{x^5} = 3 \times x^{1 - 5} = 3x^{-4} $, and then rewrite as $ \frac{3}{x^4} $ (since $ x^{-4} = \frac{1}{x^4} $)
# Answer: $ \frac{3}{x^4} $

### Problem 4: $ \frac{2yx^3}{3x^2y^3} $
# Explanation:
## Step 1: Apply exponent rule for division on $ x $ terms
$ \frac{x^3}{x^2} = x^{3 - 2} = x^1 = x $
## Step 2: Apply exponent rule for division on $ y $ terms
$ \frac{y}{y^3} = y^{1 - 3} = y^{-2} $
## Step 3: The coefficients remain as $ \frac{2}{3} $
## Step 4: Combine terms and rewrite $ y^{-2} $ with positive exponent
Multiply the coefficient, $ x $ term, and $ y $ term: $ \frac{2}{3} \times x \times y^{-2} = \frac{2x}{3y^2} $ (since $ y^{-2} = \frac{1}{y^2} $)
# Answer: $ \frac{2x}{3y^2} $ (Wait, let's recheck. Wait, the numerator is $ 2yx^3 $ and denominator is $ 3x^2y^3 $. Let's correct that.

### Correcting Problem 4: $ \frac{2yx^3}{3x^2y^3} $
# Explanation:
## Step 1: Apply exponent rule for division on $ x $ terms
$ \frac{x^3}{x^2} = x^{3 - 2} = x $
## Step 2: Apply exponent rule for division on $ y $ terms
$ \frac{y}{y^3} = y^{1 - 3} = y^{-2} $
## Step 3: The coefficients are $ \frac{2}{3} $
## Step 4: Combine all terms
Multiply the coefficient, $ x $ term, and $ y $ term: $ \frac{2}{3} \times x \times y^{-2} $
## Step 5: Rewrite $ y^{-2} $ as $ \frac{1}{y^2} $ to have positive exponents
$ \frac{2}{3} \times x \times \frac{1}{y^2} = \frac{2x}{3y^2} $
Wait, but let's do it step by step with correct numerator and denominator.

Numerator: $ 2yx^3 = 2 \times y \times x^3 $

Denominator: $ 3x^2y^3 = 3 \times x^2 \times y^3 $

So, $ \frac{2yx^3}{3x^2y^3} = \frac{2}{3} \times \frac{y}{y^3} \times \frac{x^3}{x^2} $

For $ \frac{y}{y^3} = y^{1 - 3} = y^{-2} = \frac{1}{y^2} $

For $ \frac{x^3}{x^2} = x^{3 - 2} = x $

So, $ \frac{2}{3} \times x \times \frac{1}{y^2} = \frac{2x}{3y^2} $

# Answer: $ \frac{2x}{3y^2} $

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( 3b^2 \cdot 3b^2 \cdot 2b \)

Step 1: Multiply coefficients

Step 2: Apply exponent rule for multiplication (\( a^m \cdot a^n = a^{m + n} \))

Step 3: Combine results

Step 1: Multiply coefficients

Step 2: Apply exponent rule for multiplication on \( y \) terms

Step 3: The \( x \) term remains as \( x^3 \) (since there's only one \( x \) term)

Step 4: Combine all terms

Step 1: Apply exponent rule for division (\( \frac{a^m}{a^n} = a^{m - n} \))

Step 2: Rewrite with positive exponent (\( a^{-n} = \frac{1}{a^n} \)), but we can also simplify the coefficient and the \( x \) term directly

Answer:

Problem 2: \( y^2 \cdot 2x^3y^3 \)