Calculate (2/9)^7: Evaluating Powers of Fractions

Q: Why does the exponent apply to both the numerator and denominator?

When you raise a fraction to a power, you're multiplying the entire fraction by itself that many times. Since $ \frac{2}{9} \times \frac{2}{9} = \frac{2 \times 2}{9 \times 9} = \frac{2^2}{9^2} $, the pattern continues for any exponent!

Q: Do I need to calculate the actual numbers like 2^7 and 9^7?

Not always! The question asks for the expression, so $ \frac{2^7}{9^7} $ is the complete answer. Only calculate the actual values if specifically asked for the numerical result.

Q: What if the fraction has a negative exponent?

A negative exponent means flip and make positive! So $ \left(\frac{2}{9}\right)^{-7} = \left(\frac{9}{2}\right)^7 = \frac{9^7}{2^7} $.

Q: Can I simplify this fraction further?

In this case, 2 and 9 share no common factors (2 is prime, 9 = 3²), so $ \frac{2^7}{9^7} $ is already in simplest form as an expression.

Q: Is there a shortcut for this type of problem?

Yes! Remember the pattern: fraction to a power = numerator to that power over denominator to that power. Just apply the exponent to both parts separately!

Fraction Powers with Exponential Rules

Insert the corresponding expression:

$\left(\frac{2}{9}\right)^7=$

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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 a power (N)

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

00:11 We will apply this formula to our exercise

00:17 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:

$\left(\frac{2}{9}\right)^7=$

Step-by-step solution

To solve this problem, we'll use the rule for powers of a fraction, which states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Given the expression $\left(\frac{2}{9}\right)^7$ , we apply this exponent rule:

$\left(\frac{2}{9}\right)^7 = \frac{2^7}{9^7}$

This means we raise the numerator, 2, to the power of 7, and the denominator, 9, also to the power of 7.

The matching choice in the given options is:

Choice 1: $\frac{2^7}{9^7}$

Therefore, the solution to the problem is $\frac{2^7}{9^7}$ .

Final Answer

$\frac{2^7}{9^7}$

Key Points to Remember

Essential concepts to master this topic

Rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ applies exponent to both parts
Technique: $\left(\frac{2}{9}\right)^7 = \frac{2^7}{9^7} = \frac{128}{4782969}$
Check: Verify numerator and denominator have same exponent: $2^7$ over $9^7$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only numerator or denominator
Don't write $\left(\frac{2}{9}\right)^7 = \frac{2^7}{9}$ = wrong structure! This ignores the power rule and gives an incorrect form. Always raise both the numerator AND denominator to the same power.

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 numerator and denominator?

When you raise a fraction to a power, you're multiplying the entire fraction by itself that many times. Since $\frac{2}{9} \times \frac{2}{9} = \frac{2 \times 2}{9 \times 9} = \frac{2^2}{9^2}$ , the pattern continues for any exponent!

Do I need to calculate the actual numbers like 2^7 and 9^7?

Not always! The question asks for the expression, so $\frac{2^7}{9^7}$ is the complete answer. Only calculate the actual values if specifically asked for the numerical result.

What if the fraction has a negative exponent?

A negative exponent means flip and make positive! So $\left(\frac{2}{9}\right)^{-7} = \left(\frac{9}{2}\right)^7 = \frac{9^7}{2^7}$ .

Can I simplify this fraction further?

In this case, 2 and 9 share no common factors (2 is prime, 9 = 3²), so $\frac{2^7}{9^7}$ is already in simplest form as an expression.

Is there a shortcut for this type of problem?

Yes! Remember the pattern: fraction to a power = numerator to that power over denominator to that power. Just apply the exponent to both parts separately!

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