exponent rules practice
simplify. your answer should contain only positive exponents.
1) $n^3 \cdot n$
2) $n \cdot n \cdot n$
3) $4x^2 \cdot 4x^2$
4) $4y^4 \cdot 3x^3$
5) $x^2y^2 \cdot -3y^0$
6) $-3mn \cdot -m^2n^2$
7) $(-r)^3$
8) $(-2x)^2$
9) $(2x^4)^4$
10) $(nm^4)^4$
11) $(3m)^3$
12) $(uv^0)^2$

Question

Accepted Answer

### Problem 1: $ n^3 \cdot n $
# Explanation:
## Step1: Recall exponent rule $ a^m \cdot a^n = a^{m + n} $, here $ a = n $, $ m = 3 $, $ n = 1 $ (since $ n = n^1 $)
$ n^3 \cdot n = n^{3 + 1} $
## Step2: Simplify the exponent
$ n^{3 + 1} = n^4 $
# Answer: $ n^4 $

### Problem 2: $ n \cdot n \cdot n $
# Explanation:
## Step1: Recall exponent rule $ a^m \cdot a^n \cdot a^p = a^{m + n + p} $, here $ a = n $, $ m = n = p = 1 $ (since $ n = n^1 $)
$ n \cdot n \cdot n = n^{1 + 1 + 1} $
## Step2: Simplify the exponent
$ n^{1 + 1 + 1} = n^3 $
# Answer: $ n^3 $

### Problem 3: $ 4x^2 \cdot 4x^2 $
# Explanation:
## Step1: Multiply the coefficients and use exponent rule $ a^m \cdot a^n = a^{m + n} $ for variables
First, multiply coefficients: $ 4 \cdot 4 = 16 $
Then, for $ x $: $ x^2 \cdot x^2 = x^{2 + 2} $
## Step2: Simplify the exponent of $ x $
$ x^{2 + 2} = x^4 $, so combine with coefficient: $ 16x^4 $
# Answer: $ 16x^4 $

### Problem 4: $ 4y^4 \cdot 3x^3 $
# Explanation:
## Step1: Multiply the coefficients and keep variables with their exponents (since $ y $ and $ x $ are different variables)
Multiply coefficients: $ 4 \cdot 3 = 12 $
Variables: $ y^4 $ and $ x^3 $ remain as is (since no like bases to combine exponents)
## Step2: Combine coefficient and variables
$ 12x^3y^4 $ (order of variables can be changed, but usually alphabetical)
# Answer: $ 12x^3y^4 $

### Problem 5: $ x^2y^2 \cdot -3y^0 $
# Explanation:
## Step1: Recall $ a^0 = 1 $ (for $ a
eq 0 $), so $ y^0 = 1 $
Then, multiply coefficients and use exponent rule for $ y $: $ y^2 \cdot y^0 = y^{2 + 0} $
Coefficient: $ 1 \cdot (-3) = -3 $ (since $ x^2y^2 $ has coefficient 1)
## Step2: Simplify exponent of $ y $ and combine
$ y^{2 + 0} = y^2 $, so $ -3x^2y^2 $ (note: original attempt had a mistake, $ y^0 = 1 $, not 1 leading to miscalculation, correct is $ x^2y^2 \cdot -3y^0 = -3x^2y^{2 + 0} = -3x^2y^2 $)
# Answer: $ -3x^2y^2 $

### Problem 6: $ -3mn \cdot -m^2n^2 $
# Explanation:
## Step1: Multiply the coefficients and use exponent rule $ a^m \cdot a^n = a^{m + n} $ for each variable
Multiply coefficients: $ (-3) \cdot (-1) = 3 $ (wait, original is $ -3mn \cdot -m^2n^2 $, so coefficients: $ -3 \cdot -1 = 3 $? Wait, no: $ -3 \cdot -m^2n^2 $: the second term's coefficient is -1 (since $ -m^2n^2 = -1 \cdot m^2n^2 $)
So coefficients: $ -3 \cdot -1 = 3 $
For $ m $: $ m^1 \cdot m^2 = m^{1 + 2} $ (since $ mn = m^1n^1 $)
For $ n $: $ n^1 \cdot n^2 = n^{1 + 2} $
## Step2: Simplify exponents
$ m^{1 + 2} = m^3 $, $ n^{1 + 2} = n^3 $, so combine: $ 3m^3n^3 $
# Answer: $ 3m^3n^3 $

### Problem 7: $ (-r)^3 $
# Explanation:
## Step1: Recall $ (ab)^n = a^n b^n $, here $ a = -1 $, $ b = r $, $ n = 3 $
So $ (-r)^3 = (-1)^3 \cdot r^3 $
## Step2: Simplify $ (-1)^3 $
$ (-1)^3 = -1 $, so $ -1 \cdot r^3 = -r^3 $
# Answer: $ -r^3 $

### Problem 8: $ (-2x)^2 $
# Explanation:
## Step1: Use $ (ab)^n = a^n b^n $, here $ a = -2 $, $ b = x $, $ n = 2 $
$ (-2x)^2 = (-2)^2 \cdot x^2 $
## Step2: Simplify $ (-2)^2 $
$ (-2)^2 = 4 $, so $ 4x^2 $
# Answer: $ 4x^2 $

### Problem 9: $ (2x^4)^4 $
# Explanation:
## Step1: Use $ (ab)^n = a^n b^n $ and $ (a^m)^n = a^{m \cdot n} $
First, for coefficient: $ 2^4 $
For variable: $ (x^4)^4 = x^{4 \cdot 4} $
## Step2: Calculate coefficient and exponent
$ 2^4 = 16 $, $ x^{4 \cdot 4} = x^{16} $, so combine: $ 16x^{16} $
# Answer: $ 16x^{16} $

### Problem 10: $ (nm^4)^4 $
# Explanation:
## Step1: Use $ (ab)^n = a^n b^n $ and $ (a^m)^n = a^{m \cdot n} $ for each variable
For $ n $: $ n^1 $ becomes $ n^{1 \cdot 4} $
For $ m $: $ (m^4)^4 = m^{4 \cdot 4} $
## Step2: Simplify exponents
$ n^{1 \cdot 4} = n^4 $, $ m^{4 \cdot 4} = m^{16} $, so $ n^4m^{16} $ (or $ m^{16}n^4 $)
# Answer: $ n^4m^{16} $

### Problem 11: $ (3m)^3 $
# Explanation:
## Step1: Use $ (ab)^n = a^n b^n $, here $ a = 3 $, $ b = m $, $ n = 3 $
$ (3m)^3 = 3^3 \cdot m^3 $
## Step2: Calculate $ 3^3 $
$ 3^3 = 27 $, so $ 27m^3 $
# Answer: $ 27m^3 $

### Problem 12: $ (uv^0)^2 $
# Explanation:
## Step1: Recall $ a^0 = 1 $ (for $ a
eq 0 $), so $ v^0 = 1 $, then use $ (ab)^n = a^n b^n $
First, $ v^0 = 1 $, so $ uv^0 = u \cdot 1 = u $
Then, $ (u)^2 = u^2 $ (alternatively, use exponent rules directly: $ (uv^0)^2 = u^2 \cdot (v^0)^2 = u^2 \cdot v^{0 \cdot 2} = u^2 \cdot v^0 = u^2 \cdot 1 = u^2 $)
## Step2: Simplify
Result is $ u^2 $
# Answer: $ u^2 $

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QUESTION IMAGE

Question

Explanation:

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

Step1: Recall exponent rule \( a^m \cdot a^n = a^{m + n} \), here \( a = n \), \( m = 3 \), \( n = 1 \) (since \( n = n^1 \))

Step2: Simplify the exponent

Step1: Recall exponent rule \( a^m \cdot a^n \cdot a^p = a^{m + n + p} \), here \( a = n \), \( m = n = p = 1 \) (since \( n = n^1 \))

Step2: Simplify the exponent

Step1: Multiply the coefficients and use exponent rule \( a^m \cdot a^n = a^{m + n} \) for variables

Step2: Simplify the exponent of \( x \)

Answer:

Problem 2: \( n \cdot n \cdot n \)