Write using powers of 2, 3, and/or 5: (9^k)/(10)

To express \(\frac{9^k}{10}\) using powers of \(2\), \(3\), and/or \(5\), let's start by rewriting the numbers \(9\) and \(10\) as products of their prime factors.

- \(9 = 3^2\)
- \(10 = 2 \times 5\)

Now, substitute these into the expression:

\[
\frac{9^k}{10} = \frac{(3^2)^k}{2 \times 5}
\]

Simplify the expression:

\[
\frac{3^{2k}}{2 \times 5}
\]

We can separate this fraction into two separate fractions:

\[
3^{2k} \times \frac{1}{2 \times 5}
\]

Now, write \(\frac{1}{2 \times 5}\) as:

\[
\frac{1}{2} \times \frac{1}{5} = 2^{-1} \times 5^{-1}
\]

Putting everything together:

\[
3^{2k} \times 2^{-1} \times 5^{-1}
\]

So, the expression \(\frac{9^k}{10}\) using powers of \(2\), \(3\), and \(5\) is:

\[
3^{2k} \cdot 2^{-1} \cdot 5^{-1}
\]

Would you like further details or have any questions?

Here are 5 related questions to further expand on this concept:

1. How do you express other numbers like \(27\) or \(50\) using powers of \(2\), \(3\), and \(5\)?
2. How can we simplify expressions with multiple prime factors in the denominator?
3. How do you handle expressions with negative exponents?
4. What are the general rules for converting any number to a product of its prime factors?
5. How would you approach solving equations involving powers of prime numbers?

**Tip:** When working with exponents and prime bases, remember that each prime number's exponent is independent, allowing for direct multiplication or division.

Learn how to express the fraction 9^k / 10 using powers of 2, 3, and 5 with a detailed step-by-step solution. This problem covers key concepts in exponents and prime factorization, showing how to manipulate fractions with prime bases. Suitable for students in Grades 7-9.

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