Solve: 8²⁵ × 7²⁵ × 10²⁵ × 5²⁵ Expression with Common Exponents

Q: Why can I combine the bases when the exponents are the same?

This follows the reverse power rule! When you have $ a^m \times b^m $, it's the same as multiplying a by itself m times and b by itself m times, which equals $ (a \times b)^m $.

Q: What if the exponents were different, like 8²⁵ × 7²⁶?

You cannot combine bases when exponents are different! The rule $ a^m \times b^m = (a \times b)^m $ only works when the exponents are identical.

Q: Should I calculate 8 × 7 × 10 × 5 first?

Not necessary! The question asks for the equivalent expression, not the final numerical answer. Keep it as $ (8 \times 7 \times 10 \times 5)^{25} $ unless specifically asked to evaluate.

Q: How do I remember when to use this rule?

Look for the same exponent on different bases! If you see identical powers like $ x^5 \times y^5 \times z^5 $, you can always combine the bases: $ (xyz)^5 $.

Q: Does order matter when combining the bases?

No! Multiplication is commutative, so $ (8 \times 7 \times 10 \times 5)^{25} $ equals $ (5 \times 10 \times 7 \times 8)^{25} $. Any arrangement of the bases gives the same result.

Exponent Rules with Common Powers

Insert the corresponding expression:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a product that is raised to a power (N)

00:08 Equals a product where each factor is raised to that same power (N)

00:12 This formula is valid regardless of how many factors are in the product

00:19 We will apply this formula to our exercise

00:22 We will break down the product into each factor separately raised to the power (N)

00:34 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

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

Step-by-step solution

To solve this problem, we'll apply the power of a product rule to the given expression:

Step 1: Recognize that each term $8^{25}$ , $7^{25}$ , $10^{25}$ , and $5^{25}$ is raised to the same power of 25.

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

Step 3: Combine the expression: $(8 \times 7 \times 10 \times 5)^{25}$ .

Therefore, the simplified expression is $\left(8 \times 7 \times 10 \times 5\right)^{25}$ .

This matches choice 1 from the provided options.

Final Answer

$\left(8\times7\times10\times5\right)^{25}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When multiplying terms with same exponents, combine bases first
Technique: $a^m \times b^m = (a \times b)^m$ applies to multiple terms
Check: Verify all terms have identical exponent 25 before combining ✓

Common Mistakes

Avoid these frequent errors

Trying to add exponents instead of combining bases
Don't write $8^{25} \times 7^{25} = 8^{50} \times 7^{50}$ = completely wrong result! Adding exponents only works when bases are the same. Always combine the bases first when exponents are identical: $(8 \times 7 \times 10 \times 5)^{25}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine the bases when the exponents are the same?

This follows the reverse power rule! When you have $a^m \times b^m$ , it's the same as multiplying a by itself m times and b by itself m times, which equals $(a \times b)^m$ .

What if the exponents were different, like 8²⁵ × 7²⁶?

You cannot combine bases when exponents are different! The rule $a^m \times b^m = (a \times b)^m$ only works when the exponents are identical.

Should I calculate 8 × 7 × 10 × 5 first?

Not necessary! The question asks for the equivalent expression, not the final numerical answer. Keep it as $(8 \times 7 \times 10 \times 5)^{25}$ unless specifically asked to evaluate.

How do I remember when to use this rule?

Look for the same exponent on different bases! If you see identical powers like $x^5 \times y^5 \times z^5$ , you can always combine the bases: $(xyz)^5$ .

Does order matter when combining the bases?

No! Multiplication is commutative, so $(8 \times 7 \times 10 \times 5)^{25}$ equals $(5 \times 10 \times 7 \times 8)^{25}$ . Any arrangement of the bases gives the same result.

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