Question
use product rule of exponents to simplify the expression 2*(4) . 2*(2)
Answers
GPT-4o mini
To simplify the expression \( 2^{4} \cdot 2^{2} \) using the product rule of exponents, we apply the rule that states:
\[
a^m \cdot a^n = a^{m+n}
\]
In this case, we have:
\[
2^{4} \cdot 2^{2} = 2^{4+2} = 2^{6}
\]
Thus, the simplified expression is:
\[
2^{6}
\]
If you want the numerical value, \( 2^{6} = 64 \).
So, the final simplified result is:
\[
2^{6} \quad \text{or} \quad 64
\]
\[
a^m \cdot a^n = a^{m+n}
\]
In this case, we have:
\[
2^{4} \cdot 2^{2} = 2^{4+2} = 2^{6}
\]
Thus, the simplified expression is:
\[
2^{6}
\]
If you want the numerical value, \( 2^{6} = 64 \).
So, the final simplified result is:
\[
2^{6} \quad \text{or} \quad 64
\]
use product rule of exponents to simplify the expression 3*(8) . 3*(3)
GPT-4o mini
To simplify the expression \( 3^{8} \cdot 3^{3} \) using the product rule of exponents, we apply the rule that states:
\[
a^m \cdot a^n = a^{m+n}
\]
In this case, we have:
\[
3^{8} \cdot 3^{3} = 3^{8+3} = 3^{11}
\]
Thus, the simplified expression is:
\[
3^{11}
\]
If you want the numerical value, \( 3^{11} = 177147 \) (calculated using \( 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)).
So, the final simplified result is:
\[
3^{11} \quad \text{or} \quad 177147
\]
\[
a^m \cdot a^n = a^{m+n}
\]
In this case, we have:
\[
3^{8} \cdot 3^{3} = 3^{8+3} = 3^{11}
\]
Thus, the simplified expression is:
\[
3^{11}
\]
If you want the numerical value, \( 3^{11} = 177147 \) (calculated using \( 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)).
So, the final simplified result is:
\[
3^{11} \quad \text{or} \quad 177147
\]
use product rule of exponents to simplify the expression 7*(5) . 7 *(2)
GPT-4o mini
To simplify the expression \( 7^{5} \cdot 7^{2} \) using the product rule of exponents, we apply the rule that states:
\[
a^m \cdot a^n = a^{m+n}
\]
In this case, we have:
\[
7^{5} \cdot 7^{2} = 7^{5+2} = 7^{7}
\]
Thus, the simplified expression is:
\[
7^{7}
\]
If you want the numerical value, \( 7^{7} = 823543 \) (calculated using \( 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \)).
So, the final simplified result is:
\[
7^{7} \quad \text{or} \quad 823543
\]
\[
a^m \cdot a^n = a^{m+n}
\]
In this case, we have:
\[
7^{5} \cdot 7^{2} = 7^{5+2} = 7^{7}
\]
Thus, the simplified expression is:
\[
7^{7}
\]
If you want the numerical value, \( 7^{7} = 823543 \) (calculated using \( 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \)).
So, the final simplified result is:
\[
7^{7} \quad \text{or} \quad 823543
\]
find the numerical equivalent of 7 *(2). 7*(3)
GPT-4o mini
To find the numerical equivalent of \( 7^{2} \cdot 7^{3} \), we first apply the product rule of exponents:
\[
7^{2} \cdot 7^{3} = 7^{2+3} = 7^{5}
\]
Now, we need to calculate \( 7^{5} \):
\[
7^{5} = 7 \times 7 \times 7 \times 7 \times 7
\]
Calculating this step by step:
\[
7 \times 7 = 49
\]
\[
49 \times 7 = 343
\]
\[
343 \times 7 = 2401
\]
\[
2401 \times 7 = 16807
\]
So, the numerical equivalent of \( 7^{2} \cdot 7^{3} \) is:
\[
7^{5} = 16807
\]
Thus, the final answer is \( 16807 \).
\[
7^{2} \cdot 7^{3} = 7^{2+3} = 7^{5}
\]
Now, we need to calculate \( 7^{5} \):
\[
7^{5} = 7 \times 7 \times 7 \times 7 \times 7
\]
Calculating this step by step:
\[
7 \times 7 = 49
\]
\[
49 \times 7 = 343
\]
\[
343 \times 7 = 2401
\]
\[
2401 \times 7 = 16807
\]
So, the numerical equivalent of \( 7^{2} \cdot 7^{3} \) is:
\[
7^{5} = 16807
\]
Thus, the final answer is \( 16807 \).
find the numerical equivalent of 3 *(9), 3 *(-5)
GPT-4o mini
To find the numerical equivalent of \( 3^{9} \) and \( 3^{-5} \), we can calculate each term separately:
1. **Calculating \( 3^{9} \)**:
\[
3^{9} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3
\]
Using intermediate calculations:
- \( 3^2 = 9 \)
- \( 3^3 = 27 \)
- \( 3^4 = 81 \)
- \( 3^5 = 243 \)
- \( 3^6 = 729 \)
- \( 3^7 = 2187 \)
- \( 3^8 = 6561 \)
- \( 3^9 = 19683 \)
Thus, \( 3^{9} = 19683 \).
2. **Calculating \( 3^{-5} \)**:
- The expression \( 3^{-5} \) represents the reciprocal of \( 3^{5} \):
\[
3^{-5} = \frac{1}{3^{5}} = \frac{1}{243}
\]
(since \( 3^{5} = 243 \)).
So, the numerical equivalents are:
- \( 3^{9} = 19683 \)
- \( 3^{-5} = \frac{1}{243} \) or approximately \( 0.004115226 \).
If you would like to combine both terms in a specific way, please provide further details!
1. **Calculating \( 3^{9} \)**:
\[
3^{9} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3
\]
Using intermediate calculations:
- \( 3^2 = 9 \)
- \( 3^3 = 27 \)
- \( 3^4 = 81 \)
- \( 3^5 = 243 \)
- \( 3^6 = 729 \)
- \( 3^7 = 2187 \)
- \( 3^8 = 6561 \)
- \( 3^9 = 19683 \)
Thus, \( 3^{9} = 19683 \).
2. **Calculating \( 3^{-5} \)**:
- The expression \( 3^{-5} \) represents the reciprocal of \( 3^{5} \):
\[
3^{-5} = \frac{1}{3^{5}} = \frac{1}{243}
\]
(since \( 3^{5} = 243 \)).
So, the numerical equivalents are:
- \( 3^{9} = 19683 \)
- \( 3^{-5} = \frac{1}{243} \) or approximately \( 0.004115226 \).
If you would like to combine both terms in a specific way, please provide further details!