Solve (2×4×5)/7 Raised to Power a: Complete the Expression

Q: Why can't I just keep (2×4×5)^a as one group in the numerator?

You can write it that way, but it's not fully simplified! The exponent rule $ (abc)^n = a^n \times b^n \times c^n $ lets you distribute the power to each factor separately for a cleaner expression.

Q: Do I apply the exponent to the denominator too?

Yes! When you have $ \left(\frac{a}{b}\right)^n $, the exponent applies to both top and bottom: $ \frac{a^n}{b^n} $. This is the quotient rule for exponents.

Q: What's the difference between the first and second answer choices?

The first choice $ \frac{(2×4×5)^a}{7} $ forgot to apply the exponent to the denominator 7. The correct second choice $ \frac{2^a×4^a×5^a}{7^a} $ properly applies the exponent to everything.

Q: Is there a shortcut for remembering this rule?

Think "exponents are greedy" - they want to attach to every single factor! When you see $ (\text{anything})^a $, that exponent a goes on every piece inside the parentheses.

Q: Can I simplify 2×4×5 to 40 first, then raise to power a?

You could write $ \frac{40^a}{7^a} $, but the question asks for the distributed form showing each original factor with its own exponent. Both are mathematically correct!

Exponent Rules with Fraction Distribution

Insert the corresponding expression:

$\left(\frac{2\times4\times5}{7}\right)^a=$

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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 According to the laws of exponents, a fraction raised to the power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:12 We will apply this formula to our exercise

00:21 According to the laws of exponents when a product is raised to the power (N)

00:24 it is equal to each factor in the product separately raised to the same power (N)

00:30 We will apply this formula to our exercise

00:41 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:

$\left(\frac{2\times4\times5}{7}\right)^a=$

Step-by-step solution

To solve this problem, we'll apply the exponent rule for fractions and products.

The given expression is $\left(\frac{2 \times 4 \times 5}{7}\right)^a$ . We need to simplify this using exponent rules.
First, apply the exponent to both the numerator $2 \times 4 \times 5$ and the denominator $7$ :
$\left(\frac{2 \times 4 \times 5}{7}\right)^a = \frac{(2 \times 4 \times 5)^a}{7^a}$ .
Now, apply the rule $(ab)^n = a^n \times b^n$ to distribute the exponent on the numerator:
$(2 \times 4 \times 5)^a = 2^a \times 4^a \times 5^a$ .
Therefore, the expression simplifies to:
$\frac{2^a \times 4^a \times 5^a}{7^a}$ .

Therefore, the expression simplifies to $\frac{2^a \times 4^a \times 5^a}{7^a}$ .

Final Answer

$\frac{2^a\times4^a\times5^a}{7^a}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising fractions to powers, exponent applies to numerator and denominator
Distribution: For products like (2×4×5)^a, each factor gets exponent: 2^a×4^a×5^a
Check: Verify that both numerator and denominator have correct exponents ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to the entire numerator
Don't write (2×4×5)^a as just one grouped term in the numerator = missing the distribution rule! This ignores that each factor needs its own exponent. Always distribute the exponent to each individual factor: 2^a×4^a×5^a.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why can't I just keep (2×4×5)^a as one group in the numerator?

You can write it that way, but it's not fully simplified! The exponent rule $(abc)^n = a^n \times b^n \times c^n$ lets you distribute the power to each factor separately for a cleaner expression.

Do I apply the exponent to the denominator too?

Yes! When you have $\left(\frac{a}{b}\right)^n$ , the exponent applies to both top and bottom: $\frac{a^n}{b^n}$ . This is the quotient rule for exponents.

What's the difference between the first and second answer choices?

The first choice $\frac{(2×4×5)^a}{7}$ forgot to apply the exponent to the denominator 7. The correct second choice $\frac{2^a×4^a×5^a}{7^a}$ properly applies the exponent to everything.

Is there a shortcut for remembering this rule?

Think "exponents are greedy" - they want to attach to every single factor! When you see $(\text{anything})^a$ , that exponent a goes on every piece inside the parentheses.

Can I simplify 2×4×5 to 40 first, then raise to power a?

You could write $\frac{40^a}{7^a}$ , but the question asks for the distributed form showing each original factor with its own exponent. Both are mathematically correct!

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