Evaluate (5/y)^7: Simplifying a Fraction Raised to the 7th Power

Q: Why does the exponent apply to both parts of the fraction?

Think of $ \left(\frac{5}{y}\right)^7 $ as multiplying the fraction by itself 7 times. Each time you multiply, both the 5 and the y get multiplied, so both need the exponent!

Q: Do I need to calculate 5^7?

Not necessarily! The answer $ \frac{5^7}{y^7} $ is perfectly acceptable. However, if needed, $ 5^7 = 78,125 $, so you could write $ \frac{78,125}{y^7} $.

Q: What if the denominator has a coefficient, like (5/3y)^7?

The same rule applies! $ \left(\frac{5}{3y}\right)^7 = \frac{5^7}{(3y)^7} = \frac{5^7}{3^7y^7} $. The exponent goes to everything in both numerator and denominator.

Q: How is this different from (5y)^7?

Very different! $ (5y)^7 = 5^7y^7 $ (multiplication), while $ \left(\frac{5}{y}\right)^7 = \frac{5^7}{y^7} $ (division). The fraction bar makes a huge difference!

Q: Can I simplify this further?

Usually $ \frac{5^7}{y^7} $ is the final answer. You could also write it as $ \left(\frac{5}{y}\right)^7 $, but the expanded form is typically preferred for showing your work.

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{5}{y}\right)^7=$

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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)

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

00:11 We will apply this formula to our exercise

00:14 We'll raise both the numerator and the denominator to the power (N)

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:

$\left(\frac{5}{y}\right)^7=$

Step-by-step solution

To solve this problem and transform the expression $\left(\frac{5}{y}\right)^7$ , we need to utilize the exponent rule for powers of fractions:

Step 1: Recognize that the expression $\left(\frac{5}{y}\right)^7$ involves both the numerator 5 and the denominator $y$ .
Step 2: According to the exponent rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ , we can apply the exponent of 7 to both the numerator and the denominator.
Step 3: Applying this rule gives us $\frac{5^7}{y^7}$ . This step distributes the power to each component of the fraction, preserving the structure of the expression.

Thus, the simplified form of the expression $\left(\frac{5}{y}\right)^7$ is $\frac{5^7}{y^7}$ .

This matches choice 3 from the provided options.

Final Answer

$\frac{5^7}{y^7}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ transforms $\left(\frac{5}{y}\right)^7$ to $\frac{5^7}{y^7}$
Check: Verify both numerator and denominator have the same exponent ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only numerator or denominator
Don't write $\left(\frac{5}{y}\right)^7 = \frac{5^7}{y}$ or $\frac{5}{y^7}$ ! This breaks the exponent rule and gives incomplete results. Always apply the exponent to both the numerator AND denominator: $\frac{5^7}{y^7}$ .

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does the exponent apply to both parts of the fraction?

Think of $\left(\frac{5}{y}\right)^7$ as multiplying the fraction by itself 7 times. Each time you multiply, both the 5 and the y get multiplied, so both need the exponent!

Do I need to calculate 5^7?

Not necessarily! The answer $\frac{5^7}{y^7}$ is perfectly acceptable. However, if needed, $5^7 = 78,125$ , so you could write $\frac{78,125}{y^7}$ .

What if the denominator has a coefficient, like (5/3y)^7?

The same rule applies! $\left(\frac{5}{3y}\right)^7 = \frac{5^7}{(3y)^7} = \frac{5^7}{3^7y^7}$ . The exponent goes to everything in both numerator and denominator.

How is this different from (5y)^7?

Very different! $(5y)^7 = 5^7y^7$ (multiplication), while $\left(\frac{5}{y}\right)^7 = \frac{5^7}{y^7}$ (division). The fraction bar makes a huge difference!

Can I simplify this further?

Usually $\frac{5^7}{y^7}$ is the final answer. You could also write it as $\left(\frac{5}{y}\right)^7$ , but the expanded form is typically preferred for showing your work.

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