Evaluate (5×6)^9: Solving a Multiple-Step Exponent Problem

Q: Why can't I just apply the exponent to one of the numbers?

The exponent applies to the entire product inside the parentheses! Think of it as multiplying $ (5 \times 6) $ by itself 9 times, which means both 5 and 6 get raised to the 9th power.

Q: What's the difference between (5×6)⁹ and 5×6⁹?

$ (5 \times 6)^9 = 30^9 $ is much larger than $ 5 \times 6^9 $! The parentheses make all the difference - they show that the exponent applies to the entire product.

Q: Can I simplify 5×6 first, then raise to the 9th power?

Yes! You could calculate $ 30^9 $, but that's a huge number. It's often easier to keep it as $ 5^9 \times 6^9 $ for further calculations.

Q: Does this rule work with more than two factors?

Absolutely! For example, $ (2 \times 3 \times 4)^5 = 2^5 \times 3^5 \times 4^5 $. The exponent distributes to every single factor inside the parentheses.

Q: How do I remember this rule?

Think of it as the "sharing rule" - the exponent outside the parentheses gets shared equally with every factor inside. Each factor gets its own copy of the exponent!

Power of Product with Exponent Distribution

Choose the expression(s) that corresponds to the following:

$\left(5\times6\right)^9=$

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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 In order to open parentheses containing a multiplication exercise with an outer exponent

00:06 Raise each factor to the power

00:11 We'll apply this formula to our exercise

00:18 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression(s) that corresponds to the following:

$\left(5\times6\right)^9=$

Step-by-step solution

To solve the expression $\left(5\times6\right)^9$ , we need to apply the rule for the power of a product. This rule states that when you raise a product to a power, it is equivalent to raising each factor in the product to the same power. Mathematically, this is expressed as:

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

In this specific problem, the factors are $5$ and $6$ and the exponent is $9$ . Applying the power of a product rule, we get:

$\left(5\times6\right)^9 = 5^9 \times 6^9$

Therefore, the correct expression that corresponds to $\left(5\times6\right)^9$ is $5^9\times6^9$ .

Final Answer

$5^9\times6^9$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, distribute the exponent
Technique: $(5 \times 6)^9 = 5^9 \times 6^9$ by applying the rule
Check: Both sides equal same value when calculated: $30^9 = 5^9 \times 6^9$ ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to one factor
Don't write $(5 \times 6)^9 = 5 \times 6^9$ or $5^9 \times 6$ = wrong answer! This violates the power of a product rule and gives completely different values. Always distribute the exponent to every factor inside the parentheses.

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 apply the exponent to one of the numbers?

The exponent applies to the entire product inside the parentheses! Think of it as multiplying $(5 \times 6)$ by itself 9 times, which means both 5 and 6 get raised to the 9th power.

What's the difference between (5×6)⁹ and 5×6⁹?

$(5 \times 6)^9 = 30^9$ is much larger than $5 \times 6^9$ ! The parentheses make all the difference - they show that the exponent applies to the entire product.

Can I simplify 5×6 first, then raise to the 9th power?

Yes! You could calculate $30^9$ , but that's a huge number. It's often easier to keep it as $5^9 \times 6^9$ for further calculations.

Does this rule work with more than two factors?

Absolutely! For example, $(2 \times 3 \times 4)^5 = 2^5 \times 3^5 \times 4^5$ . The exponent distributes to every single factor inside the parentheses.

How do I remember this rule?

Think of it as the "sharing rule" - the exponent outside the parentheses gets shared equally with every factor inside. Each factor gets its own copy of the exponent!

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