Question

find the value of x.
$10^x \cdot 10^{-9} = 10^{11}$

1. write an equivalent expression for $6^2 \cdot 6^4$. show your steps.

bonus 1
what value of x makes the equation true?
$2^3 \cdot 2^x = 2^9$

Explanation:

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

Step1: Apply exponent product rule

When multiplying exponents with the same base, we add the exponents: \( a^m \cdot a^n = a^{m + n} \). So for \( 10^x \cdot 10^{-9} \), this becomes \( 10^{x + (-9)} = 10^{x - 9} \).

Step2: Set exponents equal

Since the bases are the same (\( 10 \)) and the equation \( 10^{x - 9} = 10^{11} \) holds, the exponents must be equal. So we set up the equation \( x - 9 = 11 \).

Step3: Solve for \( x \)

Add \( 9 \) to both sides of the equation: \( x = 11 + 9 = 20 \).

Step1: Recall exponent product rule

For exponents with the same base \( a^m \cdot a^n = a^{m + n} \). Here, \( a = 6 \), \( m = 2 \), and \( n = 4 \).

Step2: Apply the rule

Add the exponents: \( 6^2 \cdot 6^4 = 6^{2 + 4} = 6^6 \).

Step1: Apply exponent product rule

Using \( a^m \cdot a^n = a^{m + n} \), for \( 2^3 \cdot 2^x \), we get \( 2^{3 + x} \).

Step2: Set exponents equal

Since \( 2^{3 + x} = 2^9 \), the exponents must be equal: \( 3 + x = 9 \).

Step3: Solve for \( x \)

Subtract \( 3 \) from both sides: \( x = 9 - 3 = 6 \).

Answer:

\( x = 20 \)

Second Problem: Write an equivalent expression for \( 6^2 \cdot 6^4 \)