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

$112^0=\text{?}$

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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