Power of a Product: Combination of different bases

Given $10^{12}$ and $9^{12}$, apply the property $a^m \times b^m = (a \times b)^m$, to rewrite part of the expression as:

$10^{12} \times 9^{12} = (10 \times 9)^{12}$.

The expression now becomes:

$(10 \times 9)^{12} \times 9^6$.

Therefore, the expression $10^{12} \times 9^6 \times 9^{12}$ simplifies to $(10 \times 9)^{12} \times 9^6$.

The goal is to express the given expression $8^7 \times 8^8 \times 9^7$ using properties of exponents.

First, observe that $8^7$ and $9^7$ share a common exponent of $7$. So, they can be factored as:

$(8 \times 9)^7$.

This handles the product $8^7 \times 9^7$. Now, include $8^8$ which is not part of the factoring:

$(8 \times 9)^7 \times 8^8$.

This resulting expression matches the provided possible choice.

Therefore, the rewritten expression is $\left(8 \times 9\right)^7 \times 8^8$.

Let's simplify the expression $8^{25} \times 7^3 \times 10^3 \times 5^{25} \times 5$.

Firstly, take note of the terms that we can combine based on their exponents:

• Combine $8^{25}$ and $5^{25}$: Using the property $a^m \times b^m = (a \times b)^m$, we have:
  $8^{25} \times 5^{25} = (8 \times 5)^{25}$.
• The terms $7^3$ and $10^3$ can be combined similarly: $7^3 \times 10^3 = (7 \times 10)^3$.
• Remain aware of the remaining factor of $5$ which does not pair with others.

Putting these together, the expression can be rewritten as:

$(8 \times 5)^{25} \times (7 \times 10)^3 \times 5$

The expression is now fully simplified using the rules of exponents and the indicated product combinations.

Thus, the correct rewritten form of the expression is:

$\left(8\times5\right)^{25}\times\left(7\times10\right)^3\times5$

Let's solve the problem by following these steps:

Step 1: Identify terms that share a common power. We have $13^2$, $4^2$, and $9^2$, all raised to 2.

Step 2: Use the power of a product rule: $(a \times b \times c)^m = a^m \times b^m \times c^m$.

Step 3: Combine these terms: $13^2 \times 4^2 \times 9^2 = (13 \times 4 \times 9)^2$.

Step 4: Substitute back into the original expression:
$7^6 \times 13^2 \times 4^2 \times 8^7 \times 9^2 = 7^6 \times (13 \times 4 \times 9)^2 \times 8^7$.

Therefore, the expression reduces to $7^6 \times (13 \times 4 \times 9)^2 \times 8^7$.

To solve this problem, we'll follow these steps:

• Step 1: Group terms with the same exponent to utilize the power of a product rule.
• Step 2: Combine terms under each group using the power of a product property.
• Step 3: Match the result to one of the multiple-choice options provided.

Now, let's work through each step:
Step 1: Observe the expression $3^9 \times 12^4 \times 6^9 \times 4^9 \times 4^4 \times 7^9$. Group the terms with exponents of 9: $3^9, 6^9, 4^9,$ and $7^9$. Group those with exponents of 4: $12^4$ and $4^4$.

Step 2: For the terms with exponent 9, apply the power of a product rule: $(3 \times 6 \times 4 \times 7)^9$

For the terms with exponent 4, apply the power of a product rule: $(12 \times 4)^4$

Step 3: Combine these to form the expression: $(3 \times 6 \times 4 \times 7)^9 \times (12 \times 4)^4$

Therefore, the solution to the problem is $\left(3 \times 6 \times 4 \times 7\right)^9 \times \left(12 \times 4\right)^4$. This corresponds to choice 3.