simplify:
1. $3\cdot 4^{3}$
2. $4x^{3}\cdot 2x^{3}$
3. $x^{5}\cdot x^{3}$
4. $2x^{3}\cdot 2x^{2}$
5. $\frac{6^{5}}{6^{3}}$
6. $\frac{x^{4}}{x^{7}}$
7. $8^{0}$
8. $-(9x)^{0}$
9. $(y^{4})^{3}$
10. $(x^{2}y)^{4}$
11. $\frac{6x^{7}}{2x^{4}}$
12. $\frac{8x^{5}}{4x^{2}}$
13. $(2cd^{4})^{2}(cd)^{5}$
14. $(2fg^{4})^{4}(fg)^{6}$

Question

Accepted Answer

### Problem 1: $ 3 \cdot 4^3 $
# Explanation:
## Step 1: Calculate $ 4^3 $
$ 4^3 = 4 \times 4 \times 4 = 64 $
## Step 2: Multiply by 3
$ 3 \times 64 = 192 $
# Answer: $ 192 $

### Problem 2: $ 4x^3 \cdot 2x^3 $
# Explanation:
## Step 1: Multiply the coefficients
$ 4 \times 2 = 8 $
## Step 2: Multiply the variables (using $ a^m \cdot a^n = a^{m + n} $)
$ x^3 \cdot x^3 = x^{3 + 3} = x^6 $
## Step 3: Combine results
$ 8x^6 $
# Answer: $ 8x^6 $

### Problem 3: $ x^5 \cdot x^3 $
# Explanation:
## Step 1: Apply exponent rule $ a^m \cdot a^n = a^{m + n} $
$ x^5 \cdot x^3 = x^{5 + 3} = x^8 $
# Answer: $ x^8 $

### Problem 4: $ 2x^3 \cdot 2x^2 $
# Explanation:
## Step 1: Multiply coefficients
$ 2 \times 2 = 4 $
## Step 2: Multiply variables (exponent rule)
$ x^3 \cdot x^2 = x^{3 + 2} = x^5 $
## Step 3: Combine
$ 4x^5 $
# Answer: $ 4x^5 $

### Problem 5: $ \frac{6^5}{6^3} $
# Explanation:
## Step 1: Apply exponent rule $ \frac{a^m}{a^n} = a^{m - n} $
$ \frac{6^5}{6^3} = 6^{5 - 3} = 6^2 $
## Step 2: Calculate $ 6^2 $
$ 6^2 = 36 $
# Answer: $ 36 $

### Problem 6: $ \frac{x^4}{x^7} $
# Explanation:
## Step 1: Apply exponent rule $ \frac{a^m}{a^n} = a^{m - n} $
$ \frac{x^4}{x^7} = x^{4 - 7} = x^{-3} $
## Step 2: Rewrite with positive exponent ($ a^{-n} = \frac{1}{a^n} $)
$ x^{-3} = \frac{1}{x^3} $
# Answer: $ \frac{1}{x^3} $

### Problem 7: $ 8^0 $
# Explanation:
## Step 1: Apply exponent rule $ a^0 = 1 $ (for $ a
eq 0 $)
$ 8^0 = 1 $
# Answer: $ 1 $

### Problem 8: $ -(9x)^0 $
# Explanation:
## Step 1: Apply exponent rule $ a^0 = 1 $ (for $ a
eq 0 $)
$ (9x)^0 = 1 $ (since $ 9x
eq 0 $ is assumed)
## Step 2: Apply the negative sign
$ -1 \times 1 = -1 $
# Answer: $ -1 $

### Problem 9: $ (y^4)^3 $
# Explanation:
## Step 1: Apply exponent rule $ (a^m)^n = a^{m \times n} $
$ (y^4)^3 = y^{4 \times 3} = y^{12} $
# Answer: $ y^{12} $

### Problem 10: $ (x^2 y)^4 $
# Explanation:
## Step 1: Apply exponent rule $ (ab)^n = a^n b^n $
$ (x^2 y)^4 = (x^2)^4 \cdot y^4 $
## Step 2: Apply $ (a^m)^n = a^{m \times n} $ to $ (x^2)^4 $
$ (x^2)^4 = x^{2 \times 4} = x^8 $
## Step 3: Combine results
$ x^8 y^4 $
# Answer: $ x^8 y^4 $

### Problem 11: $ \frac{6x^7}{2x^4} $
# Explanation:
## Step 1: Divide coefficients
$ \frac{6}{2} = 3 $
## Step 2: Divide variables (exponent rule $ \frac{a^m}{a^n} = a^{m - n} $)
$ \frac{x^7}{x^4} = x^{7 - 4} = x^3 $
## Step 3: Combine results
$ 3x^3 $
# Answer: $ 3x^3 $

### Problem 12: $ \frac{8x^5}{4x^2} $
# Explanation:
## Step 1: Divide coefficients
$ \frac{8}{4} = 2 $
## Step 2: Divide variables (exponent rule)
$ \frac{x^5}{x^2} = x^{5 - 2} = x^3 $
## Step 3: Combine results
$ 2x^3 $
# Answer: $ 2x^3 $

### Problem 13: $ (2cd^4)^2 (cd)^5 $
# Explanation:
## Step 1: Expand $ (2cd^4)^2 $ using $ (ab)^n = a^n b^n $ and $ (a^m)^n = a^{m \times n} $
$ (2cd^4)^2 = 2^2 \cdot c^2 \cdot (d^4)^2 = 4c^2 d^8 $
## Step 2: Expand $ (cd)^5 $ using $ (ab)^n = a^n b^n $
$ (cd)^5 = c^5 d^5 $
## Step 3: Multiply the two results
$ 4c^2 d^8 \cdot c^5 d^5 $
## Step 4: Multiply coefficients and variables (exponent rule for same base)
- Coefficients: $ 4 $
- $ c $: $ c^2 \cdot c^5 = c^{2 + 5} = c^7 $
- $ d $: $ d^8 \cdot d^5 = d^{8 + 5} = d^{13} $
## Step 5: Combine results
$ 4c^7 d^{13} $
# Answer: $ 4c^7 d^{13} $

### Problem 14: $ (2fg^4)^4 (fg)^6 $
# Explanation:
## Step 1: Expand $ (2fg^4)^4 $ using $ (ab)^n = a^n b^n $ and $ (a^m)^n = a^{m \times n} $
$ (2fg^4)^4 = 2^4 \cdot f^4 \cdot (g^4)^4 = 16f^4 g^{16} $
## Step 2: Expand $ (fg)^6 $ using $ (ab)^n = a^n b^n $
$ (fg)^6 = f^6 g^6 $
## Step 3: Multiply the two results
$ 16f^4 g^{16} \cdot f^6 g^6 $
## Step 4: Multiply coefficients and variables (exponent rule for same base)
- Coefficients: $ 16 $
- $ f $: $ f^4 \cdot f^6 = f^{4 + 6} = f^{10} $
- $ g $: $ g^{16} \cdot g^6 = g^{16 + 6} = g^{22} $
## Step 5: Combine results
$ 16f^{10} g^{22} $
# Answer: $ 16f^{10} g^{22} $

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( 3 \cdot 4^3 \)

Step 1: Calculate \( 4^3 \)

Step 2: Multiply by 3

Step 1: Multiply the coefficients

Step 2: Multiply the variables (using \( a^m \cdot a^n = a^{m + n} \))

Step 3: Combine results

Step 1: Apply exponent rule \( a^m \cdot a^n = a^{m + n} \)

Answer:

Problem 2: \( 4x^3 \cdot 2x^3 \)