Reduce the Expression: 8^25 × 7^3 × 10^3 × 5^25 × 5 Step-by-Step

Q: Why can't I combine all the terms into one big exponent?

The exponent rule $ a^m \times b^m = (a \times b)^m $ only works when the exponents are already the same. Since we have exponents of 25, 3, 3, 25, and 1, we can only group terms with matching exponents.

Q: What do I do with the single factor of 5?

The lone factor of 5 (which is really $ 5^1 $) doesn't match any other exponent, so it stays separate. Don't try to force it into a group!

Q: How do I know which terms to group together?

Look for identical exponents! In this problem: $ 8^{25} $ and $ 5^{25} $ both have exponent 25, while $ 7^3 $ and $ 10^3 $ both have exponent 3.

Q: Is my final answer fully simplified?

Yes! $ (8 \times 5)^{25} \times (7 \times 10)^3 \times 5 $ is the most simplified form using exponent rules. You could calculate the products inside the parentheses, but that's not required for this type of problem.

Q: Can I use this same method for different exponents?

Absolutely! This grouping technique works whenever you have terms with matching exponents. Just remember: only combine terms that share the exact same exponent value.

Exponent Rules with Same Base Grouping

Reduce the following equation:

$8^{25}\times7^3\times10^3\times5^{25}\times5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 Any number raised to the power of 1 is always equal to itself

00:11 According to the laws of exponents, a product raised to the power (N)

00:14 equals the product where each factor is raised to the same power (N)

00:18 Identify all numbers with equal exponents

00:22 Apply the formula to our exercise

00:28 Identify all numbers with equal exponents and apply the formula

00:35 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$8^{25}\times7^3\times10^3\times5^{25}\times5=$

Step-by-step solution

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:

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Same Exponent Rule: $a^m \times b^m = (a \times b)^m$ combines terms with matching powers
Grouping Strategy: Pair $8^{25} \times 5^{25} = (8 \times 5)^{25}$ and $7^3 \times 10^3 = (7 \times 10)^3$
Final Check: Count all factors: 5 terms become 3 groups with no terms lost ✓

Common Mistakes

Avoid these frequent errors

Combining all terms with one common exponent
Don't write $(8 \times 7 \times 10 \times 5 \times 5)^{25}$ = wrong grouping! This forces all terms to have the same exponent when they don't. Always group only terms that already share the same exponent, leaving unmatched terms separate.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I combine all the terms into one big exponent?

The exponent rule $a^m \times b^m = (a \times b)^m$ only works when the exponents are already the same. Since we have exponents of 25, 3, 3, 25, and 1, we can only group terms with matching exponents.

What do I do with the single factor of 5?

The lone factor of 5 (which is really $5^1$ ) doesn't match any other exponent, so it stays separate. Don't try to force it into a group!

How do I know which terms to group together?

Look for identical exponents! In this problem: $8^{25}$ and $5^{25}$ both have exponent 25, while $7^3$ and $10^3$ both have exponent 3.

Is my final answer fully simplified?

Yes! $(8 \times 5)^{25} \times (7 \times 10)^3 \times 5$ is the most simplified form using exponent rules. You could calculate the products inside the parentheses, but that's not required for this type of problem.

Can I use this same method for different exponents?

Absolutely! This grouping technique works whenever you have terms with matching exponents. Just remember: only combine terms that share the exact same exponent value.

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