Evaluate ((2×8)/(7×19))^(2x+3y): Complex Fraction Power Expression

Q: Why does the exponent go on every number in the fraction?

When you raise a fraction to a power, the entire fraction is being multiplied by itself that many times. Since $ \frac{2×8}{7×19} $ is one fraction, every part gets the exponent (2x+3y).

Q: Can I simplify the fraction first before applying the exponent?

Yes! You could calculate $ \frac{2×8}{7×19} = \frac{16}{133} $ first, then raise to the power. However, keeping it factored as shown often makes the algebra cleaner.

Q: What's the difference between the wrong answers?

Option A only applies the exponent to some factors, while Option B incorrectly splits the exponent. Only Option C correctly applies (2x+3y) to every single factor in the fraction.

Q: How do I remember which factors get the exponent?

Think of the fraction as one unit. Just like $ (abc)^n = a^n b^n c^n $, everything inside the parentheses gets the exponent - both top and bottom of the fraction.

Q: Is there a shortcut for checking my answer?

Yes! Count the exponents: you should see (2x+3y) appear exactly 4 times total - once on each of the four numbers: 2, 8, 7, and 19.

Exponent Laws with Fraction Bases

Insert the corresponding expression:

$\left(\frac{2\times8}{7\times19}\right)^{2x+3y}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Let's simplify this math problem together!

00:17 Remember, when a fraction is raised to the power of N, both the top and bottom are raised to N.

00:24 So, let's apply this rule to our problem.

00:27 Next, when a whole product is raised to the power of N, each part gets the power of N too.

00:37 Let's use this idea in our exercise.

00:43 Applying these rules makes it easier!

01:13 And that's how you solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{2\times8}{7\times19}\right)^{2x+3y}=$

Final Answer

$\frac{2^{2x+3y}\times8^{2x+3y}}{7^{2x+3y}\times19^{2x+3y}}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Raise both numerator and denominator to the exponent
Distribution: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ applies to each factor
Check: Each term in numerator and denominator gets same exponent ✓

Common Mistakes

Avoid these frequent errors

Only raising part of the fraction to the power
Don't just raise 8 and 19 to the power (2x+3y) while leaving 2 and 7 unchanged = incomplete distribution! The exponent must apply to every factor in both numerator and denominator. Always raise each factor: 2, 8, 7, and 19 all get the exponent (2x+3y).

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 go on every number in the fraction?

When you raise a fraction to a power, the entire fraction is being multiplied by itself that many times. Since $\frac{2×8}{7×19}$ is one fraction, every part gets the exponent (2x+3y).

Can I simplify the fraction first before applying the exponent?

Yes! You could calculate $\frac{2×8}{7×19} = \frac{16}{133}$ first, then raise to the power. However, keeping it factored as shown often makes the algebra cleaner.

What's the difference between the wrong answers?

Option A only applies the exponent to some factors, while Option B incorrectly splits the exponent. Only Option C correctly applies (2x+3y) to every single factor in the fraction.

How do I remember which factors get the exponent?

Think of the fraction as one unit. Just like $(abc)^n = a^n b^n c^n$ , everything inside the parentheses gets the exponent - both top and bottom of the fraction.

Is there a shortcut for checking my answer?

Yes! Count the exponents: you should see (2x+3y) appear exactly 4 times total - once on each of the four numbers: 2, 8, 7, and 19.

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