Solve: 20¹⁰ × 4¹⁰ × 2¹⁰ - Multiplication of Powers Problem

Q: Why can I combine the bases before raising to the power?

This works because of the power of a product rule: $ (abc)^n = a^n \times b^n \times c^n $. We're just using it in reverse! When you see the same exponent on multiple bases, you can group them first.

Q: What if the exponents were different numbers?

Then you cannot use this rule! For example, $ 20^8 \times 4^{10} \times 2^{12} $ cannot be simplified using this method. The exponents must be exactly the same.

Q: Do I need to calculate the final numerical answer?

Usually no! The simplified form $ (20 \times 4 \times 2)^{10} = 160^{10} $ is the complete answer. Computing 160¹⁰ would give an enormous number that's not practical to write out.

Q: Can this work with more than three numbers?

Absolutely! You can combine any number of bases as long as they all have the same exponent. For example: $ 2^5 \times 3^5 \times 4^5 \times 5^5 = (2 \times 3 \times 4 \times 5)^5 $

Q: What's the most common mistake students make?

Many students try to add the bases instead of multiplying them, or they think they should add the exponents. Remember: same exponents mean multiply the bases!

Exponent Rules with Product Combinations

Choose the expression that corresponds to the following:

$20^{10}\times4^{10}\times2^{10}=$

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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)

00:06 All of the factors can be written to the power (N)

00:11 We will apply this formula to our exercise

00:19 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:

$20^{10}\times4^{10}\times2^{10}=$

Step-by-step solution

To solve the expression $20^{10} \times 4^{10} \times 2^{10}$ , we can apply the power of a product rule for exponents. According to this rule, for any numbers $a$ , $b$ , and $c$ raised to the power of $n$ , we we can state:

$a^n \times b^n \times c^n = (a \times b \times c)^n$

In this case, we have:

$a = 20$
$b = 4$
$c = 2$
$n = 10$

Thus, we can write:

$20^{10} \times 4^{10} \times 2^{10} = (20 \times 4 \times 2)^{10}$

Final Answer

$\left(20\times4\times2\right)^{10}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases have same exponent, combine first then raise to power
Technique: $20^{10} \times 4^{10} \times 2^{10} = (20 \times 4 \times 2)^{10}$
Check: Calculate base first: 20 × 4 × 2 = 160, so answer is $160^{10}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying bases
Don't write $20^{10} \times 4^{10} \times 2^{10} = (20 + 4 + 2)^{10}$ = wrong answer! Adding gives 26¹⁰ instead of 160¹⁰, which is vastly different. Always multiply the bases when they have the same exponent.

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 before raising to the power?

This works because of the power of a product rule: $(abc)^n = a^n \times b^n \times c^n$ . We're just using it in reverse! When you see the same exponent on multiple bases, you can group them first.

What if the exponents were different numbers?

Then you cannot use this rule! For example, $20^8 \times 4^{10} \times 2^{12}$ cannot be simplified using this method. The exponents must be exactly the same.

Do I need to calculate the final numerical answer?

Usually no! The simplified form $(20 \times 4 \times 2)^{10} = 160^{10}$ is the complete answer. Computing 160¹⁰ would give an enormous number that's not practical to write out.

Can this work with more than three numbers?

Absolutely! You can combine any number of bases as long as they all have the same exponent. For example: $2^5 \times 3^5 \times 4^5 \times 5^5 = (2 \times 3 \times 4 \times 5)^5$

What's the most common mistake students make?

Many students try to add the bases instead of multiplying them, or they think they should add the exponents. Remember: same exponents mean multiply the bases!

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