Simplify (x×a)^30 / (a×x)^15: Exponential Fraction Problem

Q: Does it matter that the expression is written as (x×a) and (a×x)?

No, it doesn't matter! Multiplication is commutative, meaning (x×a) = (a×x). They represent the exact same base, so you can apply the quotient rule normally.

Q: What if the bottom exponent is larger than the top exponent?

You still subtract! For example, $ \frac{x^5}{x^8} = x^{5-8} = x^{-3} $. The negative exponent means $ \frac{1}{x^3} $.

Q: How can I remember which operation to use?

Use this memory trick: Division = Subtraction, Multiplication = Addition. When you see a fraction bar (division), think subtraction of exponents!

Q: Can I simplify the final answer further?

Yes! $ (x \times a)^{15} $ can be written as $ x^{15} \times a^{15} $ or $ a^{15}x^{15} $, but the form in the answer choices is perfectly acceptable.

Exponential Division with Same Base

Insert the corresponding expression:

$\frac{\left(x\times a\right)^{30}}{\left(a\times x\right)^{15}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 In multiplication, the order of factors doesn't matter

00:06 We will use this formula in our exercise and swap between the factors

00:10 According to exponent laws, division of powers with equal bases (A)

00:14 equals the same base (A) raised to the difference of exponents (M-N)

00:18 We will use this formula in our exercise

00:21 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Insert the corresponding expression:

$\frac{\left(x\times a\right)^{30}}{\left(a\times x\right)^{15}}=$

Step-by-step solution

The given expression is: $\frac{\left(x\times a\right)^{30}}{\left(a\times x\right)^{15}}$

To solve this, we can apply the quotient rule for exponents. The quotient rule states that $\frac{b^m}{b^n} = b^{m-n}$ , where $b$ is the base and $m$ and $n$ are the exponents.

In this problem, both the numerator and the denominator have the same base $\left(x \times a\right)$ . Thus, the expression simplifies by subtracting the exponents:

The exponent of the numerator is 30.
The exponent of the denominator is 15.

Applying the power of a quotient rule, we have:

$\left(x \times a\right)^{30-15}$

Thus, the simplified expression is $\left(x \times a\right)^{15}$ .

The solution to the question is: $\left(x \times a\right)^{15}$ .

Final Answer

$\left(x\times a\right)^{30-15}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract exponents
Technique: $\frac{b^m}{b^n} = b^{m-n}$ becomes $(x\times a)^{30-15}$
Check: Verify that $(x\times a)^{15} \times (x\times a)^{15} = (x\times a)^{30}$ ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents (30+15=45) or multiply them (30×15=450) = completely wrong operations! These rules apply to different situations, not division. Always subtract the bottom exponent from the top exponent when dividing same bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract the exponents when dividing?

When you divide powers with the same base, you're essentially canceling out repeated multiplication. Think of it as $\frac{a \times a \times ... (30 times)}{a \times a \times ... (15 times)}$ - the bottom 15 factors cancel with 15 from the top, leaving 15 factors.

Does it matter that the expression is written as (x×a) and (a×x)?

No, it doesn't matter! Multiplication is commutative, meaning (x×a) = (a×x). They represent the exact same base, so you can apply the quotient rule normally.

What if the bottom exponent is larger than the top exponent?

You still subtract! For example, $\frac{x^5}{x^8} = x^{5-8} = x^{-3}$ . The negative exponent means $\frac{1}{x^3}$ .

How can I remember which operation to use?

Use this memory trick: Division = Subtraction, Multiplication = Addition. When you see a fraction bar (division), think subtraction of exponents!

Can I simplify the final answer further?

Yes! $(x \times a)^{15}$ can be written as $x^{15} \times a^{15}$ or $a^{15}x^{15}$ , but the form in the answer choices is perfectly acceptable.

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