Question
which expression is equivalent to 6^4 * 5*4?
30^8
30^4
30^16
1/30^4
30^8
30^4
30^16
1/30^4
Answers
GPT-4o mini
Let's simplify the expression \(6^4 \times 5 \times 4\):
1. \(6^4\) can be rewritten using its prime factorization:
\[
6 = 2 \times 3 \implies 6^4 = (2 \times 3)^4 = 2^4 \times 3^4
\]
2. Now let's look at \(5 \times 4\):
\[
4 = 2^2 \implies 5 \times 4 = 5 \times 2^2
\]
3. Now we can substitute back into the expression:
\[
6^4 \times 5 \times 4 = (2^4 \times 3^4) \times (5 \times 2^2) = 2^{4+2} \times 3^4 \times 5 = 2^6 \times 3^4 \times 5
\]
4. Next, we observe that \(30\) can be expressed as:
\[
30 = 2 \times 3 \times 5
\]
So we have:
\[
30^n = (2 \times 3 \times 5)^n = 2^n \times 3^n \times 5^n
\]
5. To find \(n\) such that \(30^n = 2^6 \times 3^4 \times 5\), we need the exponents to match.
- For \(2\): \(n = 6\)
- For \(3\): \(n = 4\)
- For \(5\): \(n = 1\)
6. The highest exponent is \(6\), which suggests there's no single \(n\) that can equate \(30^n\) to our expression.
However, since finding exact equivalence seems complex, we can summarize our findings directly:
- The base \(30\) raised to any power would not render \(2^6 \times 3^4 \times 5\) an equivalent expression unless all bases align which they don't.
Given the options \(30^8\), \(30^4\), \(30^{16}\), and \(1/30^{4}\), none of them can be successfully derived from \(6^4 \times 5 \times 4\).
The problem likely wants the best match:
Thus, while none are equivalent, among the listed choices, **none of them are equal to** the expression calculated.
Should you have alternative calculations or confirmation, consider providing further insight or validating foundational steps for convergence checks on options.
1. \(6^4\) can be rewritten using its prime factorization:
\[
6 = 2 \times 3 \implies 6^4 = (2 \times 3)^4 = 2^4 \times 3^4
\]
2. Now let's look at \(5 \times 4\):
\[
4 = 2^2 \implies 5 \times 4 = 5 \times 2^2
\]
3. Now we can substitute back into the expression:
\[
6^4 \times 5 \times 4 = (2^4 \times 3^4) \times (5 \times 2^2) = 2^{4+2} \times 3^4 \times 5 = 2^6 \times 3^4 \times 5
\]
4. Next, we observe that \(30\) can be expressed as:
\[
30 = 2 \times 3 \times 5
\]
So we have:
\[
30^n = (2 \times 3 \times 5)^n = 2^n \times 3^n \times 5^n
\]
5. To find \(n\) such that \(30^n = 2^6 \times 3^4 \times 5\), we need the exponents to match.
- For \(2\): \(n = 6\)
- For \(3\): \(n = 4\)
- For \(5\): \(n = 1\)
6. The highest exponent is \(6\), which suggests there's no single \(n\) that can equate \(30^n\) to our expression.
However, since finding exact equivalence seems complex, we can summarize our findings directly:
- The base \(30\) raised to any power would not render \(2^6 \times 3^4 \times 5\) an equivalent expression unless all bases align which they don't.
Given the options \(30^8\), \(30^4\), \(30^{16}\), and \(1/30^{4}\), none of them can be successfully derived from \(6^4 \times 5 \times 4\).
The problem likely wants the best match:
Thus, while none are equivalent, among the listed choices, **none of them are equal to** the expression calculated.
Should you have alternative calculations or confirmation, consider providing further insight or validating foundational steps for convergence checks on options.