Simplify the Expression: (5^7 × 4^7) ÷ 19^7

Q: Why can I combine 5^7 and 4^7 but not the denominator?

You can combine terms in multiplication when they have the same exponent! Since $ 5^7 \times 4^7 $ are multiplied together, they become $ (5 \times 4)^7 $. The denominator $ 19^7 $ stays separate because it's not being multiplied with the numerator terms.

Q: How do I know which form is the most simplified?

Both $ \frac{(5 \times 4)^7}{19^7} $ and $ \left(\frac{5 \times 4}{19}\right)^7 $ are correct! The second form is often preferred because it shows the single power applied to the entire fraction, making it cleaner.

Q: Can I simplify 5 × 4 to 20 in my answer?

Yes, absolutely! You can write $ \left(\frac{20}{19}\right)^7 $ instead of $ \left(\frac{5 \times 4}{19}\right)^7 $. Both are mathematically equivalent and correct.

Q: What if the exponents were different numbers?

If exponents don't match, you cannot combine the bases! For example, $ 5^7 \times 4^6 $ stays as is - you can't make it $ (5 \times 4)^{something} $.

Q: Why are both B and C correct in this problem?

Both expressions are equivalent but represent different stages of simplification! $ \frac{(5 \times 4)^7}{19^7} $ shows the numerator simplified, while $ \left(\frac{5 \times 4}{19}\right)^7 $ applies the fraction power rule. They're the same value!

Exponent Properties with Same Powers

Insert the corresponding expression:

$\frac{5^7\times4^7}{19^7}=$

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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 exponent laws, a fraction raised to power (N)

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

00:14 According to exponent laws, a product raised to the power (N)

00:17 equals the product broken down into factors with each factor raised to the power (N)

00:20 We'll apply this formula to our exercise, converting parentheses with exponents

00:29 Now we'll apply the second formula and convert the expression into a fraction inside of parentheses with exponents

00:39 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:

$\frac{5^7\times4^7}{19^7}=$

Step-by-step solution

To solve this problem, we need to simplify the expression $\frac{5^7 \times 4^7}{19^7}$ .

We start by applying the property of exponents which states that for any numbers $a$ , $b$ , and $c$ , $a^c \times b^c = (a \times b)^c$ . In this case, we have:

$5^7 \times 4^7 = (5 \times 4)^7$

Thus, the original expression becomes:

$\frac{(5 \times 4)^7}{19^7}$

Now, using the exponent rule for powers of a fraction, $\left(\frac{a}{b}\right)^c = \frac{a^c}{b^c}$ , this further simplifies as:

$\left(\frac{5 \times 4}{19}\right)^7$

Therefore, when we look at the answer choices, both options:

$\frac{(5 \times 4)^7}{19^7}$
$\left(\frac{5 \times 4}{19}\right)^7$

are equivalent to our simplified expression. This corresponds to choice B and choice C in the given question. Therefore, both B and C are correct.

To conclude, the correct answer is $\text{B+C are correct}$ .

Final Answer

B+C are correct

Key Points to Remember

Essential concepts to master this topic

Rule: $a^n \times b^n = (a \times b)^n$ when exponents match
Technique: $5^7 \times 4^7 = (5 \times 4)^7 = 20^7$
Check: Verify $\frac{20^7}{19^7} = \left(\frac{20}{19}\right)^7$ using fraction rule ✓

Common Mistakes

Avoid these frequent errors

Applying exponent rules when powers don't match
Don't try to combine $5^7 \times 4^6$ into $(5 \times 4)^7$ = wrong result! This rule only works when exponents are identical. Always check that all powers match before combining bases.

Practice Quiz

Test your knowledge with interactive questions

Insert the corresponding expression:

$\left(\frac{2}{3}\right)^a=$

FAQ

Everything you need to know about this question

Why can I combine 5^7 and 4^7 but not the denominator?

You can combine terms in multiplication when they have the same exponent! Since $5^7 \times 4^7$ are multiplied together, they become $(5 \times 4)^7$ . The denominator $19^7$ stays separate because it's not being multiplied with the numerator terms.

How do I know which form is the most simplified?

Both $\frac{(5 \times 4)^7}{19^7}$ and $\left(\frac{5 \times 4}{19}\right)^7$ are correct! The second form is often preferred because it shows the single power applied to the entire fraction, making it cleaner.

Can I simplify 5 × 4 to 20 in my answer?

Yes, absolutely! You can write $\left(\frac{20}{19}\right)^7$ instead of $\left(\frac{5 \times 4}{19}\right)^7$ . Both are mathematically equivalent and correct.

What if the exponents were different numbers?

If exponents don't match, you cannot combine the bases! For example, $5^7 \times 4^6$ stays as is - you can't make it $(5 \times 4)^{something}$ .

Why are both B and C correct in this problem?

Both expressions are equivalent but represent different stages of simplification! $\frac{(5 \times 4)^7}{19^7}$ shows the numerator simplified, while $\left(\frac{5 \times 4}{19}\right)^7$ applies the fraction power rule. They're the same value!

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Simplify the Expression: (5^7 × 4^7) ÷ 19^7

Exponent Properties with Same Powers

❤️ 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 can I combine 5^7 and 4^7 but not the denominator?

How do I know which form is the most simplified?

Can I simplify 5 × 4 to 20 in my answer?

What if the exponents were different numbers?

Why are both B and C correct in this problem?

🌟 Unlock Your Math Potential

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