expand and simplify:
4. ( x^2 cdot x^5 ) 5. ( b^4 cdot b^7 ) 6. ( n^6 cdot n^2 )

simplify without expanding:
7. ( y^4 cdot y^3 ) 8. ( y^0 cdot x^9 ) 9. ( m^9 cdot m^5 )

expand and simplify:
10. ( (x^3)^2 ) 11. ( (a^2)^4 ) 12. ( (y^1)^6 )

simplify without expanding:
13. ( (x^4)^9 ) 14. ( (a^6)^4 ) 15. ( (y^7)^3 )

what is the rule for multiplying with the same base (product rule)?

what is the rule when you have a power raised to a power (power rule)?

can you write ( x^2 cdot y^5 ) as a single base and exponent? why or why not?

Question

expand and simplify:
4. ( x^2 cdot x^5 ) 5. ( b^4 cdot b^7 ) 6. ( n^6 cdot n^2 )

simplify without expanding:
7. ( y^4 cdot y^3 ) 8. ( y^0 cdot x^9 ) 9. ( m^9 cdot m^5 )

expand and simplify:
10. ( (x^3)^2 ) 11. ( (a^2)^4 ) 12. ( (y^1)^6 )

simplify without expanding:
13. ( (x^4)^9 ) 14. ( (a^6)^4 ) 15. ( (y^7)^3 )

what is the rule for multiplying with the same base (product rule)?

what is the rule when you have a power raised to a power (power rule)?

can you write ( x^2 cdot y^5 ) as a single base and exponent? why or why not?

Accepted Answer

### Problem 4: $ x^2 \cdot x^5 $
# Explanation:
## Step1: Recall Product Rule
For same base $ a $, $ a^m \cdot a^n = a^{m + n} $. Here, base is $ x $, $ m = 2 $, $ n = 5 $.
## Step2: Apply the rule
$ x^2 \cdot x^5 = x^{2 + 5} = x^7 $
# Answer: $ x^7 $

### Problem 5: $ b^4 \cdot b^7 $
# Explanation:
## Step1: Use Product Rule
Base $ b $, $ m = 4 $, $ n = 7 $. Formula: $ a^m \cdot a^n = a^{m + n} $.
## Step2: Calculate exponent
$ b^4 \cdot b^7 = b^{4 + 7} = b^{11} $
# Answer: $ b^{11} $

### Problem 6: $ n^6 \cdot n^2 $
# Explanation:
## Step1: Apply Product Rule
Base $ n $, $ m = 6 $, $ n = 2 $. Rule: $ a^m \cdot a^n = a^{m + n} $.
## Step2: Sum exponents
$ n^6 \cdot n^2 = n^{6 + 2} = n^8 $
# Answer: $ n^8 $

### Problem 7: $ y^4 \cdot y^3 $ (simplify without expanding)
# Explanation:
## Step1: Recall Product Rule
For $ a^m \cdot a^n $, result is $ a^{m + n} $. Base $ y $, $ m = 4 $, $ n = 3 $.
## Step2: Compute exponent
$ y^4 \cdot y^3 = y^{4 + 3} = y^7 $
# Answer: $ y^7 $

### Problem 8: $ y^0 \cdot x^9 $ (simplify without expanding)
# Explanation:
## Step1: Recall zero - exponent rule
Any non - zero number to the power of 0 is 1, so $ y^0 = 1 $ (assuming $ y
eq0 $).
## Step2: Multiply by $ x^9 $
$ y^0 \cdot x^9=1\cdot x^9 = x^9 $
# Answer: $ x^9 $

### Problem 9: $ m^9 \cdot m^5 $ (simplify without expanding)
# Explanation:
## Step1: Use Product Rule
Base $ m $, $ m = 9 $, $ n = 5 $. Formula $ a^m \cdot a^n=a^{m + n} $.
## Step2: Add exponents
$ m^9 \cdot m^5=m^{9 + 5}=m^{14} $
# Answer: $ m^{14} $

### Problem 10: $ (x^3)^2 $ (expand and simplify)
# Explanation:
## Step1: Recall Power Rule
For $ (a^m)^n=a^{m\times n} $. Here, $ a = x $, $ m = 3 $, $ n = 2 $.
## Step2: Apply the rule
$ (x^3)^2=x^{3\times2}=x^6 $
# Answer: $ x^6 $

### Problem 11: $ (a^2)^4 $ (expand and simplify)
# Explanation:
## Step1: Use Power Rule
$ (a^m)^n=a^{m\times n} $, where $ a = a $, $ m = 2 $, $ n = 4 $.
## Step2: Calculate exponent
$ (a^2)^4=a^{2\times4}=a^8 $
# Answer: $ a^8 $

### Problem 12: $ (y^1)^6 $ (expand and simplify)
# Explanation:
## Step1: Apply Power Rule
$ (a^m)^n=a^{m\times n} $, with $ a = y $, $ m = 1 $, $ n = 6 $.
## Step2: Compute exponent
$ (y^1)^6=y^{1\times6}=y^6 $
# Answer: $ y^6 $

### Problem 13: $ (x^4)^9 $ (simplify without expanding)
# Explanation:
## Step1: Recall Power Rule
$ (a^m)^n=a^{m\times n} $, here $ a = x $, $ m = 4 $, $ n = 9 $.
## Step2: Multiply exponents
$ (x^4)^9=x^{4\times9}=x^{36} $
# Answer: $ x^{36} $

### Problem 14: $ (a^6)^4 $ (simplify without expanding)
# Explanation:
## Step1: Use Power Rule
$ (a^m)^n=a^{m\times n} $, $ a = a $, $ m = 6 $, $ n = 4 $.
## Step2: Calculate exponent
$ (a^6)^4=a^{6\times4}=a^{24} $
# Answer: $ a^{24} $

### Problem 15: $ (y^7)^3 $ (simplify without expanding)
# Explanation:
## Step1: Apply Power Rule
$ (a^m)^n=a^{m\times n} $, $ a = y $, $ m = 7 $, $ n = 3 $.
## Step2: Multiply exponents
$ (y^7)^3=y^{7\times3}=y^{21} $
# Answer: $ y^{21} $

### Question: What is the rule for multiplying with the same base (Product Rule)?
# Brief Explanations:
When multiplying two exponential expressions with the same non - zero base $ a $ (where $ a$ is a real number, variable, or algebraic expression) and exponents $ m $ and $ n $ (which are real numbers), the Product Rule states that $ a^m\cdot a^n=a^{m + n} $. This means that we keep the same base and add the exponents. For example, $ 2^3\cdot2^4 = 2^{3 + 4}=2^7 $.
# Answer: For the same base $ a$ ( $ a
eq0$ ), $ a^m\cdot a^n=a^{m + n}$ (keep the base, add the exponents)

### Question: What is the rule when you have a power raised to a power (Power Rule)?
# Brief Explanations:
When we have an exponential expression $ (a^m) $ raised to another power $ n $ (where $ a$ is a non - zero real number, variable, or algebraic expression, and $ m$ and $ n$ are real numbers), the Power Rule states that $ (a^m)^n=a^{m\times n} $. In other words, we keep the base and multiply the exponents. For example, $ (3^2)^4 = 3^{2\times4}=3^8 $.
# Answer: For $ (a^m)^n$ ( $ a
eq0$ ), $ (a^m)^n=a^{m\times n}$ (keep the base, multiply the exponents)

### Question: Can you write $ x^2\cdot y^5 $ as a single base and exponent? Why or why not?
# Brief Explanations:
To write an expression as a single base with an exponent, the bases of the exponential terms must be the same. In the expression $ x^2\cdot y^5 $, the bases are $ x $ and $ y $, which are different (assuming $ x
eq y$ and they are not related by a power - of - a - power or product rule for the same base). The Product Rule for exponents ($ a^m\cdot a^n=a^{m + n}$) only applies when the bases $ a $ are the same. Since $ x $ and $ y $ are different bases, we cannot combine $ x^2\cdot y^5 $ into a single base with an exponent using the standard exponent rules for multiplication.
# Answer: No. Because the bases $ x $ and $ y $ are different, and the exponent product rule (for same base) or other exponent combination rules do not apply to combine different bases into one.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 4: \( x^2 \cdot x^5 \)

Step1: Recall Product Rule

Step2: Apply the rule

Step1: Use Product Rule

Step2: Calculate exponent

Step1: Apply Product Rule

Step2: Sum exponents

Answer:

Problem 5: \( b^4 \cdot b^7 \)