Simplify the Exponential Fraction: (2×9)^(x+7) ÷ (9×2)^(y+4)

Q: Why can I treat (2×9) and (9×2) as the same base?

Because multiplication is commutative! This means $ 2 \times 9 = 9 \times 2 = 18 $. Order doesn't matter in multiplication, so they represent the same base value.

Q: What if the bases looked completely different?

If the bases were truly different (like $ 3^x \div 5^y $), you cannot use the quotient rule. The quotient rule $ \frac{a^m}{a^n} = a^{m-n} $ only works when bases are identical.

Q: How do I subtract the exponents correctly?

Use parentheses! Write $ (x+7) - (y+4) $, then distribute the negative: $ x + 7 - y - 4 = x - y + 3 $. The parentheses prevent sign errors.

Q: Can I leave my answer as 18^(x-y+3)?

Check the answer choices! The problem asks you to match a specific format. Since $ 18 = 9 \times 2 $, write your answer as $ (9 \times 2)^{x-y+3} $ to match the given options.

Q: What if I expanded (2×9) to 18 but forgot about (9×2)?

You'd get confused trying to simplify $ \frac{18^{x+7}}{(9 \times 2)^{y+4}} $! Always recognize that both expressions equal 18 before applying exponent rules. This makes the problem much cleaner.

Exponent Rules with Equivalent Base Fractions

Insert the corresponding expression:

$\frac{\left(2\times9\right)^{x+7}}{\left(9\times2\right)^{y+4}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:14 Simply put,

00:17 In multiplication, order doesn't matter.

00:20 We'll practice this by switching the factors.

00:27 Now, let's look at dividing powers.

00:30 A to the power of N divided by A to the power of M,

00:35 equals A to the power of M minus N.

00:39 We'll use this idea in our exercise.

00:45 Let's carefully open the parentheses.

00:48 Remember, negative times positive is always negative.

01:00 Now, let's group the factors together.

01:04 And that is how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(2\times9\right)^{x+7}}{\left(9\times2\right)^{y+4}}=$

Step-by-step solution

Let's start solving the expression:

$\frac{\left(2\times9\right)^{x+7}}{\left(9\times2\right)^{y+4}}=$

First, observe the base of the numerators and denominators. Both are essentially equal since $2\times9 = 9\times2 = 18$ .

Thus, the expression can be written as:

$\frac{18^{x+7}}{18^{y+4}}$

According to the rule of exponents \/, where $\frac{a^m}{a^n} = a^{m-n}$ , we can subtract the exponents in such a situation:

Therefore, our expression becomes:

$18^{(x+7)-(y+4)}$

Now simplify the exponent:

$x + 7 - y - 4 = x - y + 3$

Thus, the final simplified expression is:

$18^{x-y+3}$

Observe that $18 = 9 \times 2$ , hence the expression can also be rewritten as:

$\left(9\times2\right)^{x-y+3}$

The solution to the question is: $\left(9\times2\right)^{x-y+3}$

Final Answer

$\left(9\times2\right)^{x-y+3}$

Key Points to Remember

Essential concepts to master this topic

Base Recognition: Identify that 2×9 equals 9×2 before simplifying
Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ when bases are identical
Check: Verify (x+7)-(y+4) = x-y+3 by expanding carefully ✓

Common Mistakes

Avoid these frequent errors

Treating different-looking bases as unequal
Don't assume (2×9) and (9×2) are different bases = wrong simplification! Multiplication is commutative, so 2×9 = 9×2 = 18. Always recognize equivalent bases before applying exponent rules.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I treat (2×9) and (9×2) as the same base?

Because multiplication is commutative! This means $2 \times 9 = 9 \times 2 = 18$ . Order doesn't matter in multiplication, so they represent the same base value.

What if the bases looked completely different?

If the bases were truly different (like $3^x \div 5^y$ ), you cannot use the quotient rule. The quotient rule $\frac{a^m}{a^n} = a^{m-n}$ only works when bases are identical.

How do I subtract the exponents correctly?

Use parentheses! Write $(x+7) - (y+4)$ , then distribute the negative: $x + 7 - y - 4 = x - y + 3$ . The parentheses prevent sign errors.

Can I leave my answer as 18^(x-y+3)?

Check the answer choices! The problem asks you to match a specific format. Since $18 = 9 \times 2$ , write your answer as $(9 \times 2)^{x-y+3}$ to match the given options.

What if I expanded (2×9) to 18 but forgot about (9×2)?

You'd get confused trying to simplify $\frac{18^{x+7}}{(9 \times 2)^{y+4}}$ ! Always recognize that both expressions equal 18 before applying exponent rules. This makes the problem much cleaner.

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Simplify the Exponential Fraction: (2×9)^(x+7) ÷ (9×2)^(y+4)

Exponent Rules with Equivalent Base Fractions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can I treat (2×9) and (9×2) as the same base?

What if the bases looked completely different?

How do I subtract the exponents correctly?

Can I leave my answer as 18^(x-y+3)?

What if I expanded (2×9) to 18 but forgot about (9×2)?

🌟 Unlock Your Math Potential

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