Simplify (ax)^21 / (xa)^6: Advanced Exponent Rules Practice

Q: Why are (ax) and (xa) the same thing?

Because of the commutative property of multiplication! This means a × x = x × a, so the order doesn't matter. It's like saying 3 × 5 = 5 × 3.

Q: Can I add the exponents instead of subtracting?

No! Adding exponents is for multiplication: $ a^m \times a^n = a^{m+n} $. For division, you subtract: $ \frac{a^m}{a^n} = a^{m-n} $.

Q: What if the bases looked completely different?

Then you cannot use the quotient rule! You can only combine exponents when the bases are exactly the same. Always check if bases are equivalent first.

Q: Why is the answer (xa)¹⁵ and not (ax)¹⁵?

Both are correct! Since ax = xa, you can write the answer either way. The problem just chose to match the format from the denominator.

Q: How do I remember when to add vs subtract exponents?

Think multiplication = add and division = subtract. When you see a fraction bar (÷), you subtract the bottom exponent from the top exponent.

Exponent Division with Commutative Properties

Insert the corresponding expression:

$\frac{\left(a\times x\right)^{21}}{\left(x\times a\right)^6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 The order of factors in multiplication doesn't matter

00:07 We'll use this formula in our exercise and switch between the factors

00:11 We'll use the formula for dividing exponents

00:13 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:16 equals the number (A) to the power of the difference of exponents (M-N)

00:20 We'll use this formula in our exercise

00:24 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(a\times x\right)^{21}}{\left(x\times a\right)^6}=$

Step-by-step solution

We start with the given expression:
$\frac{\left(a\times x\right)^{21}}{\left(x\times a\right)^6}$ .

Firstly, notice that the terms inside the parentheses in both the numerator and the denominator are the same: $a \times x$ , which are equivalent due to the commutative property of multiplication.

Thus, we can rewrite the expression in terms of $(ax)$ as follows:
$\frac{\left(a x\right)^{21}}{\left(a x\right)^6}$

We will apply the power of a quotient rule for exponents, which states that:

$\frac{a^m}{a^n} = a^{m-n}$ , where $a \neq 0$ .

Using this rule to simplify the expression, we have:

$\frac{(ax)^{21}}{(ax)^6} = (ax)^{21-6} = (ax)^{15}$ .

Thus, the expression simplifies to $(a \times x)^{15}$ .

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Commutative Rule: ax and xa are identical due to multiplication order
Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ gives $(ax)^{21-6} = (ax)^{15}$
Verification: Check that both parentheses contain same terms in different order ✓

Common Mistakes

Avoid these frequent errors

Treating (ax) and (xa) as different expressions
Don't think (ax)²¹ and (xa)⁶ have different bases = can't use quotient rule! This ignores that multiplication is commutative, so ax = xa. Always recognize when bases are equivalent 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 are (ax) and (xa) the same thing?

Because of the commutative property of multiplication! This means a × x = x × a, so the order doesn't matter. It's like saying 3 × 5 = 5 × 3.

Can I add the exponents instead of subtracting?

No! Adding exponents is for multiplication: $a^m \times a^n = a^{m+n}$ . For division, you subtract: $\frac{a^m}{a^n} = a^{m-n}$ .

What if the bases looked completely different?

Then you cannot use the quotient rule! You can only combine exponents when the bases are exactly the same. Always check if bases are equivalent first.

Why is the answer (xa)¹⁵ and not (ax)¹⁵?

Both are correct! Since ax = xa, you can write the answer either way. The problem just chose to match the format from the denominator.

How do I remember when to add vs subtract exponents?

Think multiplication = add and division = subtract. When you see a fraction bar (÷), you subtract the bottom exponent from the top exponent.

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