Evaluate (a·5·6·y)^5: Simplifying a Fifth Power Expression

Q: Why does the exponent apply to every factor inside the parentheses?

The power rule states that when you raise a product to a power, each factor gets raised to that power. Think of $ (a \cdot 30 \cdot y)^5 $ as multiplying $ (a \cdot 30 \cdot y) $ by itself 5 times!

Q: Should I calculate 30^5 or leave it as is?

For this type of problem, it's usually better to leave it as $ 30^5 $. Calculating $ 30^5 = 24,300,000 $ makes the answer unnecessarily complicated and harder to check.

Q: What if I have negative numbers inside the parentheses?

The same rule applies! Just remember that odd powers keep the negative sign while even powers make it positive. For example: $ (-2)^5 = -32 $.

Q: Can I multiply the constants first before applying the exponent?

Yes! It's actually helpful to simplify inside the parentheses first. So $ (a \cdot 5 \cdot 6 \cdot y)^5 = (a \cdot 30 \cdot y)^5 $, then apply the power rule.

Exponent Rules with Multiple Factors

$(a\cdot5\cdot6\cdot y)^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify

00:02 Let's calculate the possible multiplication

00:05 We'll use the power formula for multiplication

00:08 Every multiplication to the power of the exponent (N)

00:11 Equals each factor separately to the power of the same exponent (N)

00:14 We'll use this formula in our exercise

00:18 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(a\cdot5\cdot6\cdot y)^5=$

Step-by-step solution

We use the formula:

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$(a\times5\times6\times y)^5=(a\times30\times y)^5$

$a^530^5y^5$

Final Answer

$a^5\cdot30^5\cdot y^5$

Key Points to Remember

Essential concepts to master this topic

Power Rule: $(ab)^n = a^n \cdot b^n$ for all factors
Technique: First multiply constants: $5 \times 6 = 30$ , then apply exponent
Check: Each factor gets raised to the power: $a^5, 30^5, y^5$ ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to some factors
Don't raise just one factor to the 5th power like $a \cdot 30 \cdot y^5$ = wrong answer! The exponent applies to the entire product inside parentheses. Always raise every single factor to the given power.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

Why does the exponent apply to every factor inside the parentheses?

The power rule states that when you raise a product to a power, each factor gets raised to that power. Think of $(a \cdot 30 \cdot y)^5$ as multiplying $(a \cdot 30 \cdot y)$ by itself 5 times!

Should I calculate 30^5 or leave it as is?

For this type of problem, it's usually better to leave it as $30^5$ . Calculating $30^5 = 24,300,000$ makes the answer unnecessarily complicated and harder to check.

What if I have negative numbers inside the parentheses?

The same rule applies! Just remember that odd powers keep the negative sign while even powers make it positive. For example: $(-2)^5 = -32$ .

Can I multiply the constants first before applying the exponent?

Yes! It's actually helpful to simplify inside the parentheses first. So $(a \cdot 5 \cdot 6 \cdot y)^5 = (a \cdot 30 \cdot y)^5$ , then apply the power rule.

How do I check if my answer is correct?

Make sure every factor inside the original parentheses has the exponent 5. Your final answer should have three separate terms: $a^5$ , $30^5$ , and $y^5$ .

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Evaluate (a·5·6·y)^5: Simplifying a Fifth Power Expression

Exponent Rules with Multiple Factors

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does the exponent apply to every factor inside the parentheses?

Should I calculate 30^5 or leave it as is?

What if I have negative numbers inside the parentheses?

Can I multiply the constants first before applying the exponent?

How do I check if my answer is correct?

🌟 Unlock Your Math Potential

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