Simplify the Power Fraction: 7^10 ÷ 9^10

Q: Why can I combine the bases when the exponents are the same?

When you have identical exponents, the quotient rule lets you factor out the common power: $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $. Think of it as reversing the power of a quotient rule!

Q: What if the exponents were different, like 7^10 ÷ 9^8?

If exponents are different, you cannot use this rule! You'd need to calculate each power separately or use other exponent rules. The quotient rule $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $ only works when m is the same.

Q: Is (7/9)^10 really the same as the original fraction?

Yes! Both expressions represent the exact same value. $ \left(\frac{7}{9}\right)^{10} $ is just a more compact way to write $ \frac{7^{10}}{9^{10}} $.

Q: How do I remember this rule?

Think: "Same exponent, combine the bases!" When you see identical powers in a fraction, you can always move them outside as a single exponent: $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $.

Q: Can I use this rule with addition or subtraction?

No! This quotient rule only works with division (fractions). For addition like $ 7^{10} + 9^{10} $, you cannot combine the bases - you'd need to calculate each power separately.

Exponent Rules with Quotient Powers

Insert the corresponding expression:

$\frac{7^{10}}{9^{10}}=$

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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 of (N)

00:08 equals the numerator and denominator, each raised to the same power (N)

00:13 We'll apply this formula to our exercise, only this time in the opposite direction

00:20 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:

$\frac{7^{10}}{9^{10}}=$

Step-by-step solution

To solve this problem, let's transform the expression $\frac{7^{10}}{9^{10}}$ .

Step 1: Identify the Form

The expression $\frac{7^{10}}{9^{10}}$ fits the pattern $\frac{a^m}{b^m}$ .

Step 2: Apply the Power of a Quotient Rule

The power of a quotient formula is $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ .

Substitute $a = 7$ , $b = 9$ , and $m = 10$ into this formula, and we have:

$\frac{7^{10}}{9^{10}} = \left(\frac{7}{9}\right)^{10}$ .

We can see that this transformation results in the expression $\left(\frac{7}{9}\right)^{10}$ , which matches answer choice 1.

Therefore, the final expression is $\left(\frac{7}{9}\right)^{10}$ .

Thus, the correct reformulated expression is $\left(\frac{7}{9}\right)^{10}$ .

Final Answer

$\left(\frac{7}{9}\right)^{10}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same exponent, use quotient rule
Technique: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ transforms division into single power
Check: Verify $\left(\frac{7}{9}\right)^{10}$ equals original expression ✓

Common Mistakes

Avoid these frequent errors

Dividing exponents instead of applying quotient rule
Don't reduce $\frac{7^{10}}{9^{10}}$ to $\frac{7^{10}}{90}$ or $\frac{70}{9^{10}}$ ! This incorrectly applies operations to the base and exponent separately. Always use the quotient rule: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine the bases when the exponents are the same?

When you have identical exponents, the quotient rule lets you factor out the common power: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ . Think of it as reversing the power of a quotient rule!

What if the exponents were different, like 7^10 ÷ 9^8?

If exponents are different, you cannot use this rule! You'd need to calculate each power separately or use other exponent rules. The quotient rule $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ only works when m is the same.

Is (7/9)^10 really the same as the original fraction?

Yes! Both expressions represent the exact same value. $\left(\frac{7}{9}\right)^{10}$ is just a more compact way to write $\frac{7^{10}}{9^{10}}$ .

How do I remember this rule?

Think: "Same exponent, combine the bases!" When you see identical powers in a fraction, you can always move them outside as a single exponent: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ .

Can I use this rule with addition or subtraction?

No! This quotient rule only works with division (fractions). For addition like $7^{10} + 9^{10}$ , you cannot combine the bases - you'd need to calculate each power separately.

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