Calculate the Product: 9^9 × 8^9 × 2^9 Powers Multiplication

Q: Why can't I just add 9 + 8 + 2 to get the base?

That's a common confusion! The power of a product rule uses multiplication, not addition. Think of it this way: $ 2^3 \times 3^3 = (2 \times 3)^3 = 6^3 $, not $ (2 + 3)^3 = 5^3 $.

Q: When can I use this rule with different exponents?

You cannot use this rule when exponents are different! For example, $ 2^3 \times 3^4 $ cannot be simplified this way. The exponents must be exactly the same for this rule to work.

Q: What's the difference between this and adding exponents?

You add exponents when the bases are the same: $ 2^3 \times 2^4 = 2^7 $. You multiply bases when the exponents are the same: $ 2^3 \times 3^3 = (2 \times 3)^3 $.

Q: Can I calculate 9 × 8 × 2 first to check my work?

Absolutely! $ 9 \times 8 \times 2 = 144 $, so your answer becomes $ 144^9 $. This is a great way to verify you applied the rule correctly!

Q: Does the order of multiplication matter in the parentheses?

No, it doesn't! $ (9 \times 8 \times 2)^9 $ is the same as $ (2 \times 8 \times 9)^9 $ or $ (8 \times 9 \times 2)^9 $. Multiplication is commutative, so you can rearrange the factors.

Power Rules with Product Simplification

Choose the expression that corresponds to the following:

$9^9\times8^9\times2^9=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's simplify this problem step by step.

00:15 If you have a multiplication with each number raised to the same power N,

00:20 you can write power N over the whole product instead.

00:27 Let's use this pattern to solve our problem now.

00:32 And there we have our solution! Great job!

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:

$9^9\times8^9\times2^9=$

Step-by-step solution

To solve the expression $9^9 \times 8^9 \times 2^9$ , we will apply the "Power of a Product" rule in the exponent rules. This rule states that if you have a product of terms all raised to the same exponent, it can be rewritten as the product itself raised to that exponent. The formula is given by:

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

In our problem, we can identify the terms as:

$a = 9$
$b = 8$
$c = 2$
$n = 9$

Applying the formula, we can convert the product of powers into a single power:

$9^9 \times 8^9 \times 2^9 = (9 \times 8 \times 2)^9$

Final Answer

$\left(9\times8\times2\right)^9$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases are different but exponents same, combine bases first
Technique: $a^n \times b^n \times c^n = (a \times b \times c)^n$
Check: Verify by expanding: $(9 \times 8 \times 2)^9 = (144)^9$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when bases are different
Don't think $9^9 \times 8^9 \times 2^9 = (9 + 8 + 2)^9$ = wrong answer! This confuses addition with multiplication of bases. Always multiply the bases when they have the same exponent: $(9 \times 8 \times 2)^9$ .

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why can't I just add 9 + 8 + 2 to get the base?

That's a common confusion! The power of a product rule uses multiplication, not addition. Think of it this way: $2^3 \times 3^3 = (2 \times 3)^3 = 6^3$ , not $(2 + 3)^3 = 5^3$ .

When can I use this rule with different exponents?

You cannot use this rule when exponents are different! For example, $2^3 \times 3^4$ cannot be simplified this way. The exponents must be exactly the same for this rule to work.

What's the difference between this and adding exponents?

You add exponents when the bases are the same: $2^3 \times 2^4 = 2^7$ . You multiply bases when the exponents are the same: $2^3 \times 3^3 = (2 \times 3)^3$ .

Can I calculate 9 × 8 × 2 first to check my work?

Absolutely! $9 \times 8 \times 2 = 144$ , so your answer becomes $144^9$ . This is a great way to verify you applied the rule correctly!

Does the order of multiplication matter in the parentheses?

No, it doesn't! $(9 \times 8 \times 2)^9$ is the same as $(2 \times 8 \times 9)^9$ or $(8 \times 9 \times 2)^9$ . Multiplication is commutative, so you can rearrange the factors.

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