Solve (2×3)/(7×9) Raised to the 7th Power: Complete Solution

Q: Why can't I just multiply 2×3 and 7×9 first, then raise to the 7th power?

You absolutely can! $ \left(\frac{6}{63}\right)^7 = \frac{6^7}{63^7} $ gives the same result. However, the question asks for the expanded form showing all individual factors with their exponents.

Q: What's the difference between the answer choices?

Choice 1: Only applies exponent to 3 and 9, missing 2 and 7Choice 2: Treats numerator as one unit but splits denominatorChoice 3: Correctly applies exponent to all four factors

Q: Do I need to calculate the actual numerical answer?

No! The question asks for the equivalent expression, not the final number. Keep it in the form $ \frac{2^7 \times 3^7}{7^7 \times 9^7} $ to show your understanding of exponent rules.

Q: How do I remember which exponent rule to use?

Think of it in two steps: Step 1: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ (fraction rule)Step 2: $ (xy)^n = x^n y^n $ (product rule)Apply fraction rule first, then product rule to each part!

Q: What if the numbers were different?

The same rules apply! For any expression like $ \left(\frac{a \times b}{c \times d}\right)^n $, the answer is always $ \frac{a^n \times b^n}{c^n \times d^n} $. Every factor gets the exponent!

Exponent Rules with Fraction Products

Insert the corresponding expression:

$\left(\frac{2\times3}{7\times9}\right)^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to the power (N)

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

00:12 Note that both the numerator and denominator are products

00:16 We'll apply this formula to our exercise, making sure to use parentheses

00:23 According to the laws of exponents, a product raised to a power (N)

00:28 equals the product of each factor raised to that power (N)

00:32 We'll apply this formula to our exercise

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:

$\left(\frac{2\times3}{7\times9}\right)^7=$

Step-by-step solution

To solve this problem, we'll rewrite the expression using the rules of exponents.

Step 1: Identify the initial expression as $\left(\frac{2 \times 3}{7 \times 9}\right)^7$ .
Step 2: Apply the rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to distribute the exponent of 7 to both the numerator and denominator.
Step 3: Rewrite the expression as $\frac{(2 \times 3)^7}{(7 \times 9)^7}$ .
Step 4: Apply the exponent rule $(ab)^n = a^n \times b^n$ to both the numerator and the denominator.

Following these steps, we can express:

$\frac{(2 \times 3)^7}{(7 \times 9)^7} = \frac{2^7 \times 3^7}{7^7 \times 9^7}$ .

Therefore, the correct answer is $\frac{2^7 \times 3^7}{7^7 \times 9^7}$ , which corresponds to choice 3.

Final Answer

$\frac{2^7\times3^7}{7^7\times9^7}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Distribute exponents to both numerator and denominator parts
Product Rule: $(ab)^n = a^n \times b^n$ splits each factor
Check: Count factors: $2^7 \times 3^7$ in numerator, $7^7 \times 9^7$ in denominator ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only some factors
Don't write $\frac{2 \times 3^7}{7 \times 9^7}$ = wrong distribution! This violates the product rule by leaving some factors without exponents. Always apply the exponent to every single factor when distributing: $(2 \times 3)^7 = 2^7 \times 3^7$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply 2×3 and 7×9 first, then raise to the 7th power?

You absolutely can! $\left(\frac{6}{63}\right)^7 = \frac{6^7}{63^7}$ gives the same result. However, the question asks for the expanded form showing all individual factors with their exponents.

What's the difference between the answer choices?

Choice 1: Only applies exponent to 3 and 9, missing 2 and 7
Choice 2: Treats numerator as one unit but splits denominator
Choice 3: Correctly applies exponent to all four factors

Do I need to calculate the actual numerical answer?

No! The question asks for the equivalent expression, not the final number. Keep it in the form $\frac{2^7 \times 3^7}{7^7 \times 9^7}$ to show your understanding of exponent rules.

How do I remember which exponent rule to use?

Think of it in two steps:
Step 1: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (fraction rule)
Step 2: $(xy)^n = x^n y^n$ (product rule)
Apply fraction rule first, then product rule to each part!

What if the numbers were different?

The same rules apply! For any expression like $\left(\frac{a \times b}{c \times d}\right)^n$ , the answer is always $\frac{a^n \times b^n}{c^n \times d^n}$ . Every factor gets the exponent!

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