find the value of x.
$10^x \cdot 10^9 = 10^{11}$
2. 16 is \underline{\hspace{0.5cm}} times larger than 4.
3. write an equivalent expression for $6^2 \cdot 6^4$. show your steps.
4. write the product of $3 \times 10^7$ and $4 \times 10^3$ in scientific notation.
bonus 1
what value of x makes the equation true?
$2^3 \cdot 2^x = 2^9$
bonus 2
how many times larger is $10 \times 10^4$ than $2 \times 10^{12}$?
bonus 3
write an equivalent expression for $3^{-7} \cdot 3^2$. show your steps.
bonus 4
select an equivalent expression for $(6^{-3})^2$. show your steps.

Question

Accepted Answer

### Problem 1: Find the value of $ x $ in $ 10^x \cdot 10^9 = 10^{11} $
# Explanation:
## Step 1: Apply exponent product rule
When multiplying exponents with the same base, we add the exponents: $ a^m \cdot a^n = a^{m + n} $. So, $ 10^x \cdot 10^9 = 10^{x + 9} $.
## Step 2: Solve for $ x $
We have $ 10^{x + 9} = 10^{11} $. Since the bases are the same, the exponents must be equal: $ x + 9 = 11 $. Subtract 9 from both sides: $ x = 11 - 9 = 2 $.
# Answer: $ x = 2 $

### Problem 2: 16 is __ times larger than 4
# Explanation:
## Step 1: Set up division
To find how many times larger 16 is than 4, divide 16 by 4: $ \frac{16}{4} $.
## Step 2: Calculate the result
$ \frac{16}{4} = 4 $.
# Answer: 4

### Problem 3: Write an equivalent expression for $ 6^2 \cdot 6^4 $
# Explanation:
## Step 1: Apply exponent product rule
Using $ a^m \cdot a^n = a^{m + n} $, for $ 6^2 \cdot 6^4 $, add the exponents: $ 2 + 4 = 6 $.
## Step 2: Write the equivalent expression
So, $ 6^2 \cdot 6^4 = 6^{2 + 4} = 6^6 $.
# Answer: $ 6^6 $

### Problem 4: Write the product of $ 3 \times 10^7 $ and $ 4 \times 10^3 $ in scientific notation
# Explanation:
## Step 1: Multiply the coefficients and exponents separately
Multiply the coefficients: $ 3 \times 4 = 12 $. Multiply the exponents: $ 10^7 \times 10^3 = 10^{7 + 3} = 10^{10} $ (using $ a^m \cdot a^n = a^{m + n} $).
## Step 2: Adjust to scientific notation
We have $ 12 \times 10^{10} $, but scientific notation requires the coefficient to be between 1 and 10. Rewrite 12 as $ 1.2 \times 10^1 $. Then, $ 1.2 \times 10^1 \times 10^{10} = 1.2 \times 10^{1 + 10} = 1.2 \times 10^{11} $.
# Answer: $ 1.2 \times 10^{11} $

### Bonus 1: What value of $ x $ makes $ 2^3 \cdot 2^x = 2^9 $ true?
# Explanation:
## Step 1: Apply exponent product rule
$ 2^3 \cdot 2^x = 2^{3 + x} $.
## Step 2: Solve for $ x $
Set $ 3 + x = 9 $. Subtract 3 from both sides: $ x = 9 - 3 = 6 $.
# Answer: $ x = 6 $

### Bonus 2: How many times larger is $ 10 \times 10^4 $ than $ 2 \times 10^2 $?
# Explanation:
## Step 1: Simplify the numbers
First, simplify $ 10 \times 10^4 = 10^{1 + 4} = 10^5 = 100000 $. Simplify $ 2 \times 10^2 = 200 $.
## Step 2: Divide the two numbers
Divide 100000 by 200: $ \frac{100000}{200} = 500 $. Alternatively, using exponents: $ \frac{10 \times 10^4}{2 \times 10^2} = \frac{10^5}{2 \times 10^2} = \frac{1}{2} \times 10^{5 - 2} = 0.5 \times 10^3 = 5 \times 10^2 = 500 $.
# Answer: 500

### Bonus 3: Write an equivalent expression for $ 3^{-7} \cdot 3^2 $
# Explanation:
## Step 1: Apply exponent product rule
Using $ a^m \cdot a^n = a^{m + n} $, $ 3^{-7} \cdot 3^2 = 3^{-7 + 2} $.
## Step 2: Simplify the exponent
$ -7 + 2 = -5 $, so the expression is $ 3^{-5} $ (or $ \frac{1}{3^5} $, but $ 3^{-5} $ is equivalent).
# Answer: $ 3^{-5} $ (or $ \frac{1}{243} $)

### Bonus 4: Select an equivalent expression for $ (6^{-3})^2 $ (assuming the problem is $ (6^{-3})^2 $)
# Explanation:
## Step 1: Apply exponent power rule
When raising a power to a power, multiply the exponents: $ (a^m)^n = a^{m \times n} $. So, $ (6^{-3})^2 = 6^{-3 \times 2} $.
## Step 2: Simplify the exponent
$ -3 \times 2 = -6 $, so the expression is $ 6^{-6} $ (or $ \frac{1}{6^6} $).
# Answer: $ 6^{-6} $ (or $ \frac{1}{46656} $)

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QUESTION IMAGE

Question

Explanation:

Problem 1: Find the value of \( x \) in \( 10^x \cdot 10^9 = 10^{11} \)

Step 1: Apply exponent product rule

Step 2: Solve for \( x \)

Step 1: Set up division

Step 2: Calculate the result

Step 1: Apply exponent product rule

Step 2: Write the equivalent expression

Answer:

Problem 2: 16 is __ times larger than 4