Question

order the expressions by choosing \\(>\\), \\(<\\), or \\(=\\).

\\(2^2 \times 5^2 \quad \quad \quad 10^5\\)

\\(2^5 \times 2^2 \quad \quad \quad 2^{10}\\)

\\(2^5 \times 5^2 \quad \quad \quad 10^2\\)

Explanation:

Simplify the first comparison

We compare \(2^2 \times 5^2\) and \(10^5\).
Using the power of a product rule:

$$ 2^2 \times 5^2 = (2 \times 5)^2 = 10^2 $$

Since \(2 < 5\), we compare the powers of 10:

$$ 10^2 < 10^5 $$

Thus, \(2^2 \times 5^2 < 10^5\).

Simplify the second comparison

We compare \(2^5 \times 2^2\) and \(2^{10}\).
Using the product of powers rule:

$$ 2^5 \times 2^2 = 2^{5+2} = 2^7 $$

Since \(7 < 10\), we compare the powers of 2:

$$ 2^7 < 2^{10} $$

Thus, \(2^5 \times 2^2 < 2^{10}\).

Simplify the third comparison

We compare \(2^5 \times 5^2\) and \(10^2\).
We rewrite the first expression to match bases:

$$ 2^5 \times 5^2 = 2^3 \times 2^2 \times 5^2 = 2^3 \times (2 \times 5)^2 = 8 \times 10^2 $$

We compare \(8 \times 10^2\) with \(10^2\):

$$ 8 \times 10^2 > 1 \times 10^2 $$

Thus, \(2^5 \times 5^2 > 10^2\).

Answer:

No.	Problem	Answer
2	\(2^5 \times 2^2 \quad [ \quad ] \quad 2^{10}\)	\(<\)
3	\(2^5 \times 5^2 \quad [ \quad ] \quad 10^2\)	\(>\)