Solve Exponential Expression: 5^8 × 8^8 × 10^8 Multiplication

Q: Why can I only use this rule when the exponents are the same?

The power of product rule states $ (a \times b)^n = a^n \times b^n $. This only works in reverse when all exponents match. If they're different, you can't factor out a common exponent!

Q: Can I use this rule with negative exponents?

Absolutely! $ 5^{-3} \times 8^{-3} = (5 \times 8)^{-3} $. The rule works with any exponent as long as they're all the same.

Q: What's the advantage of writing it this way?

Writing $ (5 \times 8 \times 10)^8 $ instead of $ 5^8 \times 8^8 \times 10^8 $ makes calculations easier! You can multiply the bases first: $ 400^8 $ instead of calculating three separate powers.

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

Look for the same exponent on multiple terms being multiplied. If you see matching exponents, you can factor them out using this rule!

Power of Product Rule with Same Exponents

Choose the expression that corresponds to the following:

$5^8\times8^8\times10^8=$

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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 When we are presented with a multiplication operation where all the factors have the same exponent (N)

00:08 We can write the entire product with the exponent (N)

00:18 We can apply this formula to our exercise

00:26 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$5^8\times8^8\times10^8=$

Step-by-step solution

The goal is to apply the power of a product rule by transforming the expression $5^8 \times 8^8 \times 10^8$ into the form $(a \times b \times c)^n$ .

Let's begin by identifying the terms involved:

The expression consists of three separate terms, each raised to the 8th power: $5^8$ , $8^8$ , and $10^8$ .

According to the power of a product rule, $(a \times b \times c)^n = a^n \times b^n \times c^n$ . Therefore, given that the exponents are the same ( $n = 8$ ), we can reverse this process.

The original expression is $5^8 \times 8^8 \times 10^8$ .

We can then consolidate this into a single term by combining the bases under the same exponent:

Thus, $5^8 \times 8^8 \times 10^8 = (5 \times 8 \times 10)^8$ .

Therefore, the corresponding expression is:

$\left(5\times8\times10\right)^8$

Final Answer

$\left(5\times8\times10\right)^8$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases have same exponent, combine under one exponent
Technique: $5^8 \times 8^8 \times 10^8 = (5 \times 8 \times 10)^8$
Check: Verify all exponents are identical before applying the rule ✓

Common Mistakes

Avoid these frequent errors

Applying the rule when exponents don't match
Don't try to combine $5^3 \times 8^8 \times 10^8$ into $(5 \times 8 \times 10)^8$ = wrong answer! Different exponents can't be combined this way. Always check that ALL exponents are exactly the same before using the power of product rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I only use this rule when the exponents are the same?

The power of product rule states $(a \times b)^n = a^n \times b^n$ . This only works in reverse when all exponents match. If they're different, you can't factor out a common exponent!

What if I have four or more terms with the same exponent?

No problem! The rule works for any number of terms. For example: $2^5 \times 3^5 \times 4^5 \times 7^5 = (2 \times 3 \times 4 \times 7)^5$

Can I use this rule with negative exponents?

Absolutely! $5^{-3} \times 8^{-3} = (5 \times 8)^{-3}$ . The rule works with any exponent as long as they're all the same.

What's the advantage of writing it this way?

Writing $(5 \times 8 \times 10)^8$ instead of $5^8 \times 8^8 \times 10^8$ makes calculations easier! You can multiply the bases first: $400^8$ instead of calculating three separate powers.

How do I remember when to use this rule?

Look for the same exponent on multiple terms being multiplied. If you see matching exponents, you can factor them out using this rule!

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