Question
U3L10: SAMPLE WORK(1pt for correct answer and 1 pt for showing work = 10pts total) MUST SHOW HANDWRITTEN WORK FOR CREDIT. Use the Product Rule of Exponents to simplify Use the Product Rule of Exponents to simplify 6^10 x 6^5 (put answer in exponent form) 2. Find the numerical equivalent of 8^ 9 x 8^-6 3. Find the value of x to make this exponential equation true… 5^x ÷ 5^5 = 5^12 4. Use the Power Rule of Exponents to simplify (9^3)^6 . Answer in exponent form. 4. Apply the Properties of Integer Exponents to generate equivalent expressions of: (6^-3)^7 x 6^20 ━━━━━━━━ 6^-2 answer as a simplified improper fraction, even if you get a whole number answer. This is a exponent fraction problem, the line is a fraction line!
Answers
GPT-4o mini
Sure! Below are the handwritten calculations for each of the problems you provided.
### 1. Simplify \( 6^{10} \times 6^{5} \) using the Product Rule of Exponents.
The Product Rule of Exponents states that when you multiply like bases, you add the exponents.
\[
6^{10} \times 6^{5} = 6^{10 + 5} = 6^{15}
\]
### Answer:
\[ 6^{15} \]
---
### 2. Find the numerical equivalent of \( 8^{9} \times 8^{-6} \).
Using the Product Rule of Exponents, we start by simplifying:
\[
8^{9} \times 8^{-6} = 8^{9 + (-6)} = 8^{9 - 6} = 8^{3}
\]
Now, we find the numerical equivalent of \( 8^{3} \):
\[
8^{3} = 512
\]
### Answer:
\[ 512 \]
---
### 3. Find the value of \( x \) to make the equation \( \frac{5^{x}}{5^{5}} = 5^{12} \) true.
Using the Quotient Rule of Exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \):
\[
5^{x - 5} = 5^{12}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x - 5 = 12
\]
To solve for \( x \):
\[
x = 12 + 5 = 17
\]
### Answer:
\[ x = 17 \]
---
### 4. Simplify \( (9^{3})^{6} \) using the Power Rule of Exponents.
The Power Rule of Exponents states that \( (a^{m})^{n} = a^{mn} \):
\[
(9^{3})^{6} = 9^{3 \times 6} = 9^{18}
\]
### Answer:
\[ 9^{18} \]
---
### 5. Simplify \( \frac{(6^{-3})^{7} \times 6^{20}}{6^{-2}} \).
Using the Power Rule on \( (6^{-3})^{7} \):
\[
(6^{-3})^{7} = 6^{-21}
\]
Now, we can rewrite the expression:
\[
\frac{6^{-21} \times 6^{20}}{6^{-2}}
\]
Using the Product Rule in the numerator:
\[
6^{-21 + 20} = 6^{-1}
\]
Now applying the Quotient Rule:
\[
\frac{6^{-1}}{6^{-2}} = 6^{-1 - (-2)} = 6^{-1 + 2} = 6^{1}
\]
So the final simplified expression is:
### Answer:
\[ 6^{1} = 6 \]
Or in improper fraction form:
\[
\frac{6}{1}
\]
---
Feel free to let me know if you need further assistance!
### 1. Simplify \( 6^{10} \times 6^{5} \) using the Product Rule of Exponents.
The Product Rule of Exponents states that when you multiply like bases, you add the exponents.
\[
6^{10} \times 6^{5} = 6^{10 + 5} = 6^{15}
\]
### Answer:
\[ 6^{15} \]
---
### 2. Find the numerical equivalent of \( 8^{9} \times 8^{-6} \).
Using the Product Rule of Exponents, we start by simplifying:
\[
8^{9} \times 8^{-6} = 8^{9 + (-6)} = 8^{9 - 6} = 8^{3}
\]
Now, we find the numerical equivalent of \( 8^{3} \):
\[
8^{3} = 512
\]
### Answer:
\[ 512 \]
---
### 3. Find the value of \( x \) to make the equation \( \frac{5^{x}}{5^{5}} = 5^{12} \) true.
Using the Quotient Rule of Exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \):
\[
5^{x - 5} = 5^{12}
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x - 5 = 12
\]
To solve for \( x \):
\[
x = 12 + 5 = 17
\]
### Answer:
\[ x = 17 \]
---
### 4. Simplify \( (9^{3})^{6} \) using the Power Rule of Exponents.
The Power Rule of Exponents states that \( (a^{m})^{n} = a^{mn} \):
\[
(9^{3})^{6} = 9^{3 \times 6} = 9^{18}
\]
### Answer:
\[ 9^{18} \]
---
### 5. Simplify \( \frac{(6^{-3})^{7} \times 6^{20}}{6^{-2}} \).
Using the Power Rule on \( (6^{-3})^{7} \):
\[
(6^{-3})^{7} = 6^{-21}
\]
Now, we can rewrite the expression:
\[
\frac{6^{-21} \times 6^{20}}{6^{-2}}
\]
Using the Product Rule in the numerator:
\[
6^{-21 + 20} = 6^{-1}
\]
Now applying the Quotient Rule:
\[
\frac{6^{-1}}{6^{-2}} = 6^{-1 - (-2)} = 6^{-1 + 2} = 6^{1}
\]
So the final simplified expression is:
### Answer:
\[ 6^{1} = 6 \]
Or in improper fraction form:
\[
\frac{6}{1}
\]
---
Feel free to let me know if you need further assistance!