Evaluate the Expression: Simplifying (b×9×a)⁶

Q: Why does each factor get the same exponent?

The power of a product rule states that $ (xyz)^n = x^n \times y^n \times z^n $. This happens because when you multiply $ (b \times 9 \times a) $ six times, each factor appears 6 times in the multiplication!

Q: Does the order of factors matter in the final answer?

No! Multiplication is commutative, so $ b^6 \times 9^6 \times a^6 $ equals $ a^6 \times b^6 \times 9^6 $. You can arrange the factors in any order you prefer.

Q: What if one of the factors was already raised to a power?

If you had something like $ (b^2 \times 9 \times a)^6 $, you'd use the power of a power rule: $ (b^2)^6 = b^{12} $. The other factors still get the exponent 6.

Q: Can I calculate 9^6 to get a number?

You can calculate $ 9^6 = 531,441 $, but it's usually better to leave it as $ 9^6 $ unless specifically asked to evaluate. This keeps your expression cleaner and shows your understanding of the power rule.

Q: How do I remember this rule?

Think of it as distributing the exponent to each factor inside the parentheses. Just like distributing multiplication over addition, you distribute the exponent over each factor in the product!

Power Rules with Multiple Factors

Insert the corresponding expression:

$\left(b\times9\times a\right)^6=$

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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 In order to open parentheses with a multiplication operation and an outside exponent

00:09 Raise each factor to the power

00:14 We will apply this formula to our exercise

00:23 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:

$\left(b\times9\times a\right)^6=$

Step-by-step solution

To expand the expression $\left(b \times 9 \times a\right)^6$ , we'll apply the power of a product rule for exponents, which states that $(xyz)^n = x^n \times y^n \times z^n$ .

Step 1: Identify each factor within the base: $b$ , $9$ , and $a$ .
Step 2: Apply the exponent to each factor individually:

$b^6$
$9^6$
$a^6$

By multiplying these results together, the expanded form is $b^6 \times 9^6 \times a^6$ .

Therefore, the correct expression is $b^6 \times 9^6 \times a^6$ .

Final Answer

$b^6\times9^6\times a^6$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, apply the exponent to each factor
Technique: $(b \times 9 \times a)^6 = b^6 \times 9^6 \times a^6$
Check: Count factors: 3 factors in base should give 3 terms with exponent 6 ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to some factors
Don't apply the exponent 6 to just one or two factors like getting $b^6 \times 9 \times a$ = wrong result! This violates the power rule and gives an incorrect expression. Always apply the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does each factor get the same exponent?

The power of a product rule states that $(xyz)^n = x^n \times y^n \times z^n$ . This happens because when you multiply $(b \times 9 \times a)$ six times, each factor appears 6 times in the multiplication!

Does the order of factors matter in the final answer?

No! Multiplication is commutative, so $b^6 \times 9^6 \times a^6$ equals $a^6 \times b^6 \times 9^6$ . You can arrange the factors in any order you prefer.

What if one of the factors was already raised to a power?

If you had something like $(b^2 \times 9 \times a)^6$ , you'd use the power of a power rule: $(b^2)^6 = b^{12}$ . The other factors still get the exponent 6.

Can I calculate 9^6 to get a number?

You can calculate $9^6 = 531,441$ , but it's usually better to leave it as $9^6$ unless specifically asked to evaluate. This keeps your expression cleaner and shows your understanding of the power rule.

How do I remember this rule?

Think of it as distributing the exponent to each factor inside the parentheses. Just like distributing multiplication over addition, you distribute the exponent over each factor in the product!

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