Question

write each as a product of powers.

1. \\(3 \times 3 \times 3 \times 4 \times 4 = 3^{\square} \times 4^{\square}\\)
2. \\(6 \times 6 \times 6 \times 6 \times 8 \times 8 = 6^{\square} \times 8^{\square}\\)
3. \\(2.5 \times 2.5 \times 8 \times 8 = \underline{\quad\quad\quad\quad}\\)
4. \\(1.7 \times 1.7 \times 1.7 \times 1.7 \times 5 \times 5 = \underline{\quad\quad\quad\quad}\\)
5. \\(9 \times 5 \times 5 \times 9 \times 9 = \underline{\quad\quad\quad\quad}\\)
6. \\(4.6 \times 7 \times 7 \times 4.6 \times 7 = \underline{\quad\quad\quad\quad}\\)
7. \\(10 \times 7.8 \times 7.8 \times 10 \times 2 \times 2 \times 10 \times 7.8 \times 7.8 = \underline{\quad\quad\quad\quad}\\)
8. \\(6.3 \times 5 \times 5 \times 5 \times 6.3 \times 6.3 \times 13 \times 14^2 = \underline{\quad\quad\quad\quad}\\)
9. \\(3 \times 3 \times 3^2 \times 4 \times 4 \times 5.5 \times 4^3 \times 5.5 = \underline{\quad\quad\quad\quad}\\)

write the integers as powers.

1. \\(81 = 3 \times \underline{\quad\quad}\\)

\\(= 3 \times \underline{\quad\quad} \times \underline{\quad\quad}\\)
\\(= 3 \times \underline{\quad\quad} \times \underline{\quad\quad} \times \underline{\quad\quad}\\)
\\(= \underline{\quad\quad}\\)

Explanation:

Group identical factors

Count the occurrences of each unique base in the multiplication expressions.

Write as powers

Represent each repeated multiplication as a base raised to the power of its count.

Simplify expressions

Combine bases where possible using exponent rules:

1. \(3 \times 3 \times 3 \times 4 \times 4 = 3^3 \times 4^2\)
2. \(6 \times 6 \times 6 \times 6 \times 8 \times 8 = 6^4 \times 8^2\)
3. \(2.5 \times 2.5 \times 8 \times 8 = 2.5^2 \times 8^2\)
4. \(1.7 \times 1.7 \times 1.7 \times 1.7 \times 5 \times 5 = 1.7^4 \times 5^2\)
5. \(9 \times 5 \times 5 \times 9 \times 9 = 9^3 \times 5^2\)
6. \(4.6 \times 7 \times 7 \times 4.6 \times 7 = 4.6^2 \times 7^3\)
7. \(10 \times 7.8 \times 7.8 \times 10 \times 2 \times 2 \times 10 \times 7.8 \times 7.8 = 10^3 \times 7.8^4 \times 2^2\)
8. \(6.3 \times 5 \times 5 \times 5 \times 6.3 \times 6.3 \times 13 \times 14^2 = 6.3^3 \times 5^3 \times 13 \times 14^2\)
9. \(3 \times 3 \times 3^2 \times 4 \times 4 \times 5.5 \times 4^3 \times 5.5 = 3^4 \times 4^5 \times 5.5^2\)
10. \(81 = 3 \times 27 = 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 = 3^4\)

Answer:

No.	Answer
2	\(6^4 \times 8^2\)
3	\(2.5^2 \times 8^2\)
4	\(1.7^4 \times 5^2\)
5	\(9^3 \times 5^2\)
6	\(4.6^2 \times 7^3\)
7	\(10^3 \times 7.8^4 \times 2^2\)
8	\(6.3^3 \times 5^3 \times 13 \times 14^2\)
9	\(3^4 \times 4^5 \times 5.5^2\)
10	\(81 = 3 \times 27 = 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 = 3^4\)