Simplify (xa/by)⁴: Complex Fraction Power Expression

Q: Why do I need to apply the exponent to both the numerator and denominator?

The power of a quotient rule states that $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $. This means the exponent affects the entire fraction, not just part of it!

Q: What's the difference between (xa)⁴ and x⁴a⁴?

They're exactly the same! The power of a product rule tells us that $ (xa)^4 = x^4a^4 $. We just write it differently to show each variable clearly has the 4th power.

Q: Can I leave the denominator as (by)⁴ instead of expanding it?

Technically yes, but it's clearer to expand it as $ b^4y^4 $. Both forms are mathematically correct, but the expanded form shows exactly what happened to each variable.

Q: What if the original expression had different exponents on the variables?

The same rules apply! If you had $ \left(\frac{x^2a^3}{by^2}\right)^4 $, you'd get $ \frac{x^8a^{12}}{b^4y^8} $ by multiplying each existing exponent by 4.

Q: How do I check if my final answer is correct?

Look at your result and make sure every variable has been raised to the 4th power. In this case, you should see $ x^4, a^4, b^4, $ and $ y^4 $ in your answer.

Power Rule with Complex Fraction Expressions

Insert the corresponding expression:

$\left(\frac{x\times a}{b\times y}\right)^4=$

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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 a power (N)

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

00:13 We will apply this formula to our exercise

00:20 According to the laws of exponents, when a product is raised to the power (N)

00:23 it is equal to each factor in the product separately raised to the same power (N)

00:28 We will apply this formula to our exercise

00:36 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:

$\left(\frac{x\times a}{b\times y}\right)^4=$

Step-by-step solution

To solve this problem, we'll utilize the rules of exponents, particularly the rule for raising fractions to a power.

Step 1: Recognize that the expression $\left(\frac{x \times a}{b \times y}\right)^4$ has a fraction raised to a power.
Step 2: Apply the rule $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$ to distribute the exponent of 4 to both the numerator and the denominator.
Step 3: This means $\left(\frac{x \times a}{b \times y}\right)^4 = \frac{(x \times a)^4}{(b \times y)^4}$ .
Step 4: Use the power of a product rule: $(m \times n)^p = m^p \times n^p$ , which allows us to write $(x \times a)^4$ as $x^4 \times a^4$ and $(b \times y)^4$ as $b^4 \times y^4$ .
Step 5: Substitute these results back into the expression to get $\frac{x^4 \times a^4}{b^4 \times y^4}$ .

Therefore, the simplified expression is $\frac{x^4 \times a^4}{\left(b \times y\right)^4}$ .

Final Answer

$\frac{x^4\times a^4}{\left(b\times y\right)^4}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a fraction to a power, distribute the exponent to both numerator and denominator
Product Rule: Apply $(ab)^4 = a^4b^4$ to each part: $(xa)^4 = x^4a^4$
Verification: Check that all variables receive the 4th power in final answer ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to one variable in each product
Don't write $(xa)^4 = xa^4$ or $(by)^4 = by^4$ = missing powers on some variables! This ignores the power of a product rule. Always apply the exponent to every single variable: $(xa)^4 = x^4a^4$ and $(by)^4 = b^4y^4$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to both the numerator and denominator?

The power of a quotient rule states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . This means the exponent affects the entire fraction, not just part of it!

What's the difference between (xa)⁴ and x⁴a⁴?

They're exactly the same! The power of a product rule tells us that $(xa)^4 = x^4a^4$ . We just write it differently to show each variable clearly has the 4th power.

Can I leave the denominator as (by)⁴ instead of expanding it?

Technically yes, but it's clearer to expand it as $b^4y^4$ . Both forms are mathematically correct, but the expanded form shows exactly what happened to each variable.

What if the original expression had different exponents on the variables?

The same rules apply! If you had $\left(\frac{x^2a^3}{by^2}\right)^4$ , you'd get $\frac{x^8a^{12}}{b^4y^8}$ by multiplying each existing exponent by 4.

How do I check if my final answer is correct?

Look at your result and make sure every variable has been raised to the 4th power. In this case, you should see $x^4, a^4, b^4,$ and $y^4$ in your answer.

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