Question
Use the Product Rule of
Exponents to simplify
(1 point)
5 ⋅
10 5
5
25
50
5
15
25
15
5
50
Exponents to simplify
(1 point)
5 ⋅
10 5
5
25
50
5
15
25
15
5
50
Answers
Answered by
GPT-4o mini
To simplify the expression using the Product Rule of Exponents, we need to clarify the expression as a multiplication of base 5 terms and then apply the rule.
The Product Rule of Exponents states that when you multiply two expressions that have the same base, you add the exponents. The general form is:
\[ a^m \cdot a^n = a^{m+n} \]
Let's rewrite and clarify the expression, assuming it looks something like this:
\[ 5^1 \cdot 10^5 \cdot 5^{25} \cdot 5^{50} \cdot 5^{15} \cdot 5^{25} \cdot 5^{15} \cdot 5^{5} \cdot 5^{50} \]
Focusing on the powers of base 5:
1. \( 5^1 \)
2. \( 5^{25} \)
3. \( 5^{50} \)
4. \( 5^{15} \)
5. \( 5^{25} \)
6. \( 5^{15} \)
7. \( 5^{5} \)
8. \( 5^{50} \)
Now we simply sum all the exponents:
\[
1 + 25 + 50 + 15 + 25 + 15 + 5 + 50
\]
Calculating this step by step:
- \( 1 + 25 = 26 \)
- \( 26 + 50 = 76 \)
- \( 76 + 15 = 91 \)
- \( 91 + 25 = 116 \)
- \( 116 + 15 = 131 \)
- \( 131 + 5 = 136 \)
- \( 136 + 50 = 186 \)
Thus, the total exponent when combining all the \( 5^{\text{terms}} \) is \( 186 \).
So the simplified expression would be:
\[
5^{186}
\]
If you also need to attach any powers from \(10^5\) to the expression, please clarify if this was intended to be part of the multiplication or if we should just focus on the coefficients strictly involving powers of 5.
The Product Rule of Exponents states that when you multiply two expressions that have the same base, you add the exponents. The general form is:
\[ a^m \cdot a^n = a^{m+n} \]
Let's rewrite and clarify the expression, assuming it looks something like this:
\[ 5^1 \cdot 10^5 \cdot 5^{25} \cdot 5^{50} \cdot 5^{15} \cdot 5^{25} \cdot 5^{15} \cdot 5^{5} \cdot 5^{50} \]
Focusing on the powers of base 5:
1. \( 5^1 \)
2. \( 5^{25} \)
3. \( 5^{50} \)
4. \( 5^{15} \)
5. \( 5^{25} \)
6. \( 5^{15} \)
7. \( 5^{5} \)
8. \( 5^{50} \)
Now we simply sum all the exponents:
\[
1 + 25 + 50 + 15 + 25 + 15 + 5 + 50
\]
Calculating this step by step:
- \( 1 + 25 = 26 \)
- \( 26 + 50 = 76 \)
- \( 76 + 15 = 91 \)
- \( 91 + 25 = 116 \)
- \( 116 + 15 = 131 \)
- \( 131 + 5 = 136 \)
- \( 136 + 50 = 186 \)
Thus, the total exponent when combining all the \( 5^{\text{terms}} \) is \( 186 \).
So the simplified expression would be:
\[
5^{186}
\]
If you also need to attach any powers from \(10^5\) to the expression, please clarify if this was intended to be part of the multiplication or if we should just focus on the coefficients strictly involving powers of 5.