Solve for the Expression: (5×6×7/9)^(2x+1)

Q: Why do I need to apply the exponent to both numerator and denominator?

The exponent rule $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ ensures the entire fraction is raised to the power. If you only apply it to the numerator, you're changing the value of the expression!

Q: Can I distribute the exponent to each factor in the numerator?

Yes! Using $ (abc)^n = a^n \times b^n \times c^n $, you can write $ (5 \times 6 \times 7)^{2x+1} = 5^{2x+1} \times 6^{2x+1} \times 7^{2x+1} $. Both forms are equivalent!

Q: Which form is better - distributed or factored?

Both are mathematically correct! The distributed form $ \frac{5^{2x+1} \times 6^{2x+1} \times 7^{2x+1}}{9^{2x+1}} $ shows individual powers, while $ \frac{(5 \times 6 \times 7)^{2x+1}}{9^{2x+1}} $ keeps factors grouped.

Q: How do I know if my transformation is correct?

Test with simple values! Try x = 0: both forms should equal $ \frac{5 \times 6 \times 7}{9} = \frac{210}{9} $. If they match, your transformation is correct!

Q: What if the exponent has multiple terms like 2x+1?

Treat the entire expression (2x+1) as one exponent. Don't try to separate it - apply the whole thing to both numerator and denominator as a single unit.

Exponent Rules with Fraction Bases

Insert the corresponding expression:

$\left(\frac{5\times6\times7}{9}\right)^{2x+1}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's simplify this problem together.

00:15 The power law tells us that a fraction raised to the power, N, means both the top and bottom are raised to power N.

00:23 Remember, the power has an addition in it.

00:26 We'll use this rule for our problem.

00:33 If a product is raised to a power, N, each part is raised to power N separately.

00:39 Let's apply this rule to our task.

00:42 And there you have it. That's our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{5\times6\times7}{9}\right)^{2x+1}=$

Step-by-step solution

Let's solve the problem step-by-step:

We begin with the expression: $\left(\frac{5 \times 6 \times 7}{9}\right)^{2x+1}$ .

Step 1: Apply the exponent to both the numerator and the denominator using the rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

This gives: $\frac{(5 \times 6 \times 7)^{2x+1}}{9^{2x+1}}$ , as in choice b.

Step 2: Distribute the exponent across each factor in the numerator: $(a \times b \times c)^n = a^n \times b^n \times c^n$ .

This results in: $\frac{5^{2x+1} \times 6^{2x+1} \times 7^{2x+1}}{9^{2x+1}}$ .

Therefore, the expression $\left(\frac{5\times6\times7}{9}\right)^{2x+1}$ evaluates to:

$\frac{5^{2x+1} \times 6^{2x+1} \times 7^{2x+1}}{9^{2x+1}}$ .

This corresponds to choice 1. Hence, choices a' and b' are equivalent.

The correct answer is: a'+b' are correct.

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ then distribute to factors
Check: Both forms should equal same value when simplified ✓

Common Mistakes

Avoid these frequent errors

Not applying exponent to denominator
Don't just raise the numerator to the power and leave denominator unchanged = incomplete transformation! This violates the exponent rule for fractions. Always apply $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to both parts.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to both numerator and denominator?

The exponent rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ ensures the entire fraction is raised to the power. If you only apply it to the numerator, you're changing the value of the expression!

Can I distribute the exponent to each factor in the numerator?

Yes! Using $(abc)^n = a^n \times b^n \times c^n$ , you can write $(5 \times 6 \times 7)^{2x+1} = 5^{2x+1} \times 6^{2x+1} \times 7^{2x+1}$ . Both forms are equivalent!

Which form is better - distributed or factored?

Both are mathematically correct! The distributed form $\frac{5^{2x+1} \times 6^{2x+1} \times 7^{2x+1}}{9^{2x+1}}$ shows individual powers, while $\frac{(5 \times 6 \times 7)^{2x+1}}{9^{2x+1}}$ keeps factors grouped.

How do I know if my transformation is correct?

Test with simple values! Try x = 0: both forms should equal $\frac{5 \times 6 \times 7}{9} = \frac{210}{9}$ . If they match, your transformation is correct!

What if the exponent has multiple terms like 2x+1?

Treat the entire expression (2x+1) as one exponent. Don't try to separate it - apply the whole thing to both numerator and denominator as a single unit.

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