Evaluate (1/9)^7: Solving Powers of Fraction Expression

Q: What exactly is the quotient rule for exponents?

The quotient rule states that $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $. When you have the same exponent on both numerator and denominator, you can combine them into a single fraction raised to that power.

Q: Why can't I just multiply the 7 by something?

The exponent 7 applies to both 1 and 9 separately, not as a multiplier. $ \frac{1^7}{9^7} $ means 1 raised to the 7th power divided by 9 raised to the 7th power, not 7 times anything!

Q: How do I know when to use the quotient rule?

Use the quotient rule when you see the same exponent on both the numerator and denominator of a fraction. Look for the pattern $ \frac{a^n}{b^n} $ where the exponents match.

Q: Is this the same as simplifying fractions?

Not quite! This is about rewriting the form of an expression using exponent rules. We're changing $ \frac{1^7}{9^7} $ to $ \left(\frac{1}{9}\right)^7 $, which are equivalent expressions but written differently.

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

No! The question asks for the equivalent expression, not the final number. Keep it as $ \left(\frac{1}{9}\right)^7 $ unless specifically asked to calculate the decimal value.

Fraction Powers with Quotient Rule

Insert the corresponding expression:

$\frac{1^7}{9^7}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's make this problem simpler.

00:11 Remember the exponent rule: a fraction to the power of N is the same as both the top and bottom to the power of N.

00:20 We'll use this idea but in reverse for our example.

00:24 Let's apply this formula and solve it together!

00:27 And there we have our answer.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{1^7}{9^7}=$

Step-by-step solution

To solve this problem, we'll apply the formula for the power of a quotient:

Step 1: Identify the expression $\frac{1^7}{9^7}$ .
Step 2: Recognize that $\frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n$ applies here. The numerator is $a^7 = 1^7$ , and the denominator is $b^7 = 9^7$ .
Step 3: Apply the formula: $\frac{1^7}{9^7} = \left( \frac{1}{9} \right)^7$ .

In step 2, we used the property that allows us to rewrite $\frac{1^7}{9^7}$ as $\left( \frac{1}{9} \right)^7$ , which is more convenient for interpretation or further calculations.

Therefore, the expression $\frac{1^7}{9^7}$ can be rewritten as $\left( \frac{1}{9} \right)^7$ .

Final Answer

$\left(\frac{1}{9}\right)^7$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ for same exponents
Technique: Recognize $\frac{1^7}{9^7} = \left(\frac{1}{9}\right)^7$ by applying quotient rule
Check: Expand back: $\left(\frac{1}{9}\right)^7 = \frac{1^7}{9^7}$ matches original ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of applying quotient rule
Don't write $\frac{1^7}{9^7}$ as $\frac{7}{9^7}$ or multiply by 7 = completely wrong structure! This confuses the exponent with multiplication. Always use the quotient rule: $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ when both numerator and denominator have the same exponent.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

What exactly is the quotient rule for exponents?

The quotient rule states that $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ . When you have the same exponent on both numerator and denominator, you can combine them into a single fraction raised to that power.

Why can't I just multiply the 7 by something?

The exponent 7 applies to both 1 and 9 separately, not as a multiplier. $\frac{1^7}{9^7}$ means 1 raised to the 7th power divided by 9 raised to the 7th power, not 7 times anything!

How do I know when to use the quotient rule?

Use the quotient rule when you see the same exponent on both the numerator and denominator of a fraction. Look for the pattern $\frac{a^n}{b^n}$ where the exponents match.

Is this the same as simplifying fractions?

Not quite! This is about rewriting the form of an expression using exponent rules. We're changing $\frac{1^7}{9^7}$ to $\left(\frac{1}{9}\right)^7$ , which are equivalent expressions but written differently.

Do I need to calculate the actual numerical value?

No! The question asks for the equivalent expression, not the final number. Keep it as $\left(\frac{1}{9}\right)^7$ unless specifically asked to calculate the decimal value.

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