Question

1. $11^{4} cdot 11^{7} =$
2. $3^{8} cdot 3^{1} cdot 3^{4} =$
3. $\frac{10^{25}}{10^{18}} =$
4. $\frac{9^{23}}{9^{16}} =$

Explanation:

Problem 2: \( 11^4 \cdot 11^7 \)

Step 1: Apply exponent rule \( a^m \cdot a^n = a^{m + n} \)

For \( 11^4 \cdot 11^7 \), we use the rule for multiplying exponents with the same base. Here, the base \( a = 11 \), \( m = 4 \), and \( n = 7 \). So we add the exponents: \( 4 + 7 \).
\( 11^4 \cdot 11^7 = 11^{4 + 7} \)

Step 2: Calculate the sum of exponents

Calculate \( 4 + 7 = 11 \).
\( 11^{4 + 7} = 11^{11} \)

Step 1: Apply exponent rule \( a^m \cdot a^n = a^{m + n} \) (multiple times)

First, for \( 3^8 \cdot 3^1 \), the base \( a = 3 \), \( m = 8 \), \( n = 1 \). Add the exponents: \( 8 + 1 \). Then multiply the result by \( 3^4 \), using the same rule again.
\( 3^8 \cdot 3^1 \cdot 3^4 = 3^{8 + 1 + 4} \)

Step 2: Calculate the sum of exponents

Calculate \( 8 + 1 + 4 = 13 \).
\( 3^{8 + 1 + 4} = 3^{13} \)

Step 1: Apply exponent rule \( \frac{a^m}{a^n} = a^{m - n} \)

For \( \frac{10^{25}}{10^{18}} \), the base \( a = 10 \), \( m = 25 \), \( n = 18 \). So we subtract the exponents: \( 25 - 18 \).
\( \frac{10^{25}}{10^{18}} = 10^{25 - 18} \)

Step 2: Calculate the difference of exponents

Calculate \( 25 - 18 = 7 \).
\( 10^{25 - 18} = 10^7 \)

Answer:

\( 11^{11} \)

Problem 5: \( 3^8 \cdot 3^1 \cdot 3^4 \)