find the greatest common factor.
1. (4ab^{2}c^{4}-2a^{2}b^{3}c^{2}+6a^{3}b^{4}c)
2. (5m^{2}x^{2}y^{3}-10m^{3}xy^{2}+15m^{2}x^{2}y^{4})
simplify.
3. (4x^{2}(ax - 2))
4. (\frac{6a^{-3}c^{-3}}{a^{-2}cd^{0}})
5. a deck of cards contains 6 green cards and 2 yellow cards. what is the probability of drawing a green card, keeping it, and then drawing another green card?

Question

Accepted Answer

### Problem 1: Find the greatest common factor of $ 4ab^2c^4 - 2a^2b^3c^2 + 6a^3b^4c $
# Explanation:
## Step 1: Find GCF of coefficients
The coefficients are 4, -2, 6. The GCF of 4, 2, 6 is 2.
## Step 2: Find GCF of $ a $-terms
The powers of $ a $ are 1, 2, 3. The GCF is $ a^1 = a $.
## Step 3: Find GCF of $ b $-terms
The powers of $ b $ are 2, 3, 4. The GCF is $ b^2 $.
## Step 4: Find GCF of $ c $-terms
The powers of $ c $ are 4, 2, 1. The GCF is $ c^1 = c $.
## Step 5: Combine GCFs
Multiply the GCFs of coefficients, $ a $-terms, $ b $-terms, and $ c $-terms: $ 2 \times a \times b^2 \times c = 2ab^2c $.
# Answer: $ 2ab^2c $

### Problem 2: Find the greatest common factor of $ 5m^2x^2y^3 - 10m^3xy^2 + 15m^2x^2y^4 $
# Explanation:
## Step 1: Find GCF of coefficients
The coefficients are 5, -10, 15. The GCF of 5, 10, 15 is 5.
## Step 2: Find GCF of $ m $-terms
The powers of $ m $ are 2, 3, 2. The GCF is $ m^2 $.
## Step 3: Find GCF of $ x $-terms
The powers of $ x $ are 2, 1, 2. The GCF is $ x^1 = x $.
## Step 4: Find GCF of $ y $-terms
The powers of $ y $ are 3, 2, 4. The GCF is $ y^2 $.
## Step 5: Combine GCFs
Multiply the GCFs of coefficients, $ m $-terms, $ x $-terms, and $ y $-terms: $ 5 \times m^2 \times x \times y^2 = 5m^2xy^2 $.
# Answer: $ 5m^2xy^2 $

### Problem 3: Simplify $ 4x^2(ax - 2) $
# Explanation:
## Step 1: Distribute $ 4x^2 $
Use the distributive property $ a(b + c) = ab + ac $: $ 4x^2 \times ax - 4x^2 \times 2 $.
## Step 2: Multiply terms
$ 4x^2 \times ax = 4a x^{2 + 1} = 4a x^3 $ and $ 4x^2 \times 2 = 8x^2 $.
# Answer: $ 4ax^3 - 8x^2 $

### Problem 4: Simplify $ \frac{6a^{-3}c^{-3}}{a^{-2}cd^0} $
# Explanation:
## Step 1: Simplify $ d^0 $
Recall $ d^0 = 1 $ (any non - zero number to the power 0 is 1). So the expression becomes $ \frac{6a^{-3}c^{-3}}{a^{-2}c \times 1}=\frac{6a^{-3}c^{-3}}{a^{-2}c} $.
## Step 2: Use exponent rules for division ($ \frac{x^m}{x^n}=x^{m - n} $)
For $ a $-terms: $ a^{-3-(-2)}=a^{-3 + 2}=a^{-1}=\frac{1}{a} $.
For $ c $-terms: $ c^{-3-1}=c^{-4}=\frac{1}{c^4} $.
The coefficient is 6. Multiply them together: $ 6\times\frac{1}{a}\times\frac{1}{c^4}=\frac{6}{ac^4} $.
# Answer: $ \frac{6}{ac^4} $

### Problem 5: Probability of drawing two green cards (without replacement)
# Explanation:
## Step 1: Find total number of cards initially
There are 6 green cards and 2 yellow cards. So total number of cards $ N=6 + 2=8 $.
## Step 2: Probability of first green card
The probability of drawing a green card first, $ P_1=\frac{\text{Number of green cards}}{\text{Total number of cards}}=\frac{6}{8}=\frac{3}{4} $.
## Step 3: Probability of second green card (after keeping the first)
After drawing one green card, the number of green cards left is $ 6 - 1 = 5 $ and the total number of cards left is $ 8 - 1=7 $. So the probability of drawing a green card second, $ P_2=\frac{5}{7} $.
## Step 4: Probability of both events (multiplication rule for dependent events)
The probability of both events (drawing a green card and then another green card) is $ P = P_1\times P_2=\frac{3}{4}\times\frac{5}{7}=\frac{15}{28} $.
# Answer: $ \frac{15}{28} $

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QUESTION IMAGE

Question

Explanation:

Problem 1: Find the greatest common factor of \( 4ab^2c^4 - 2a^2b^3c^2 + 6a^3b^4c \)

Step 1: Find GCF of coefficients

Step 2: Find GCF of \( a \)-terms

Step 3: Find GCF of \( b \)-terms

Step 4: Find GCF of \( c \)-terms

Step 5: Combine GCFs

Step 1: Find GCF of coefficients

Step 2: Find GCF of \( m \)-terms

Step 3: Find GCF of \( x \)-terms

Step 4: Find GCF of \( y \)-terms

Step 5: Combine GCFs

Step 1: Distribute \( 4x^2 \)

Step 2: Multiply terms

Answer:

Problem 2: Find the greatest common factor of \( 5m^2x^2y^3 - 10m^3xy^2 + 15m^2x^2y^4 \)