Simplify (a/b)^9: Evaluating the Ninth Power of a Fraction

Q: Why does the exponent apply to both the top and bottom?

When you raise a fraction to a power, you're multiplying the entire fraction by itself that many times. So $ \left(\frac{a}{b}\right)^9 $ means $ \frac{a}{b} \times \frac{a}{b} \times ... $ nine times, which gives $ \frac{a^9}{b^9} $.

Q: What if the exponent was negative, like -9?

The same rule applies! $ \left(\frac{a}{b}\right)^{-9} = \frac{a^{-9}}{b^{-9}} $, which can also be written as $ \frac{b^9}{a^9} $ using the negative exponent rule.

Q: Does this work with any exponent?

Yes! The power of a fraction rule $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ works for any exponent: positive, negative, fractions, or even variables.

Q: Can I multiply the numerator and denominator first?

No! Don't try to simplify $ \frac{a}{b} $ to $ ab $ first. That changes the fraction completely. Always keep the fraction form and apply the exponent rule directly.

Q: How do I remember this rule?

Think: "The exponent visits both floors of the fraction house." It goes to both the numerator (top floor) and denominator (bottom floor), giving each the same power!

Exponent Rules with Fraction Bases

Insert the corresponding expression:

$\left(\frac{a}{b}\right)^9=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify the following problem.

00:10 Remember, when a fraction is raised to a power, like N,

00:14 both the numerator and denominator are raised to that power, N.

00:19 Now, let's apply this rule to our exercise.

00:23 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{a}{b}\right)^9=$

Step-by-step solution

The problem asks us to express $\left(\frac{a}{b}\right)^9$ using exponent rules. We will use the rule for the power of a fraction, which states:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Applying this rule, we get:

$\left(\frac{a}{b}\right)^9 = \frac{a^9}{b^9}$

This method ensures that the exponent $9$ is applied to both the numerator and the denominator of the fraction.

Therefore, the solution to the problem is $\frac{a^9}{b^9}$ .

Final Answer

$\frac{a^9}{b^9}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^9 = \frac{a^9}{b^9}$ distributes the 9 to each part
Check: Verify both parts have same exponent: a⁹ and b⁹ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only numerator or denominator
Don't write $\left(\frac{a}{b}\right)^9 = \frac{a^9}{b}$ or $\frac{a}{b^9}$ = wrong answer! The exponent must distribute to BOTH parts of the fraction. Always apply the exponent to both numerator AND denominator: $\frac{a^9}{b^9}$ .

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why does the exponent apply to both the top and bottom?

When you raise a fraction to a power, you're multiplying the entire fraction by itself that many times. So $\left(\frac{a}{b}\right)^9$ means $\frac{a}{b} \times \frac{a}{b} \times ...$ nine times, which gives $\frac{a^9}{b^9}$ .

What if the exponent was negative, like -9?

The same rule applies! $\left(\frac{a}{b}\right)^{-9} = \frac{a^{-9}}{b^{-9}}$ , which can also be written as $\frac{b^9}{a^9}$ using the negative exponent rule.

Does this work with any exponent?

Yes! The power of a fraction rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ works for any exponent: positive, negative, fractions, or even variables.

Can I multiply the numerator and denominator first?

No! Don't try to simplify $\frac{a}{b}$ to $ab$ first. That changes the fraction completely. Always keep the fraction form and apply the exponent rule directly.

How do I remember this rule?

Think: "The exponent visits both floors of the fraction house." It goes to both the numerator (top floor) and denominator (bottom floor), giving each the same power!

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