Solve the Complex Fraction: (13/126)^(x+y) Expression Completion

Q: Why do I need parentheses around the entire denominator?

The parentheses show that the entire product $ 7\times6\times3 $ gets raised to the power $ x+y $. Without them, only the last number (3) would get the exponent!

Q: What's the difference between the correct and incorrect answers?

The correct answer $ \frac{13^{x+y}}{(7\times6\times3)^{x+y}} $ applies the exponent to both parts of the fraction. The wrong answers either forget the exponent on the denominator or apply it incorrectly.

Q: Can I simplify 7×6×3 first before applying the exponent?

Yes! You can calculate $ 7\times6\times3 = 126 $ first, giving you $ \frac{13^{x+y}}{126^{x+y}} $. Both forms are mathematically equivalent.

Q: Does this rule work for any fraction raised to a power?

Absolutely! The rule $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $ works for any fraction and any exponent. It's one of the fundamental exponent laws.

Exponent Rules with Fractional Base

Insert the corresponding expression:

$\left(\frac{13}{7\times6\times3}\right)^{x+y}=$

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Understand the problem

Insert the corresponding expression:

$\left(\frac{13}{7\times6\times3}\right)^{x+y}=$

Step-by-step solution

Let's start by examining the expression given in the question:

$\left(\frac{13}{7\times6\times3}\right)^{x+y}$

This expression is a power of a fraction. There is a general rule in exponents which states:

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

Using this rule, we will apply it to our original expression.

Given, $a = 13$ , $b = 7\times6\times3$ , and $n = x+y$ , we can rewrite our expression as:

$\frac{13^{x+y}}{(7\times6\times3)^{x+y}}$

The solution to the question is:

$\frac{13^{x+y}}{(7\times6\times3)^{x+y}}$

Final Answer

$\frac{13^{x+y}}{\left(7\times6\times3\right)^{x+y}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply the power to both numerator and denominator separately
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ transforms complex fractions into manageable parts
Check: Verify denominator has parentheses around entire product: $(7\times6\times3)^{x+y}$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply exponent to entire denominator
Don't write $\frac{13^{x+y}}{7\times6\times3}$ = wrong answer! This only applies the exponent to the numerator, leaving the denominator unchanged. Always apply the exponent to both parts: $\frac{13^{x+y}}{(7\times6\times3)^{x+y}}$ .

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do I need parentheses around the entire denominator?

The parentheses show that the entire product $7\times6\times3$ gets raised to the power $x+y$ . Without them, only the last number (3) would get the exponent!

What's the difference between the correct and incorrect answers?

The correct answer $\frac{13^{x+y}}{(7\times6\times3)^{x+y}}$ applies the exponent to both parts of the fraction. The wrong answers either forget the exponent on the denominator or apply it incorrectly.

Can I simplify 7×6×3 first before applying the exponent?

Yes! You can calculate $7\times6\times3 = 126$ first, giving you $\frac{13^{x+y}}{126^{x+y}}$ . Both forms are mathematically equivalent.

Does this rule work for any fraction raised to a power?

Absolutely! The rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ works for any fraction and any exponent. It's one of the fundamental exponent laws.

What if the exponent was negative?

The same rule applies! A negative exponent would flip the fraction first, then apply the positive power. But the basic principle of applying the exponent to both parts remains unchanged.

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