Simplify (x×y)^27 ÷ (x×y)^20: Exponent Division Problem

Q: Why do we subtract exponents when dividing?

Think of it as canceling out repeated multiplication! When you have $ \frac{a^{27}}{a^{20}} $, you're canceling 20 copies of 'a' from both top and bottom, leaving $ a^{27-20} = a^7 $.

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

You still subtract! For example, $ \frac{x^3}{x^5} = x^{3-5} = x^{-2} $. The negative exponent means you have a fraction: $ \frac{1}{x^2} $.

Q: Does (x×y) count as one base or two?

It counts as one single base! Since both terms have the same base $ (x \times y) $, you can apply the division rule directly.

Q: How do I remember which operation to use?

Use this memory trick: Division = Subtraction, Multiplication = Addition. When you see a fraction bar (÷), think subtract!

Exponent Division with Same Base Terms

Insert the corresponding expression:

$\frac{\left(x\times y\right)^{27}}{\left(x\times y\right)^{20}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the formula for dividing powers

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:07 equals number (A) to the power of the difference of exponents (M-N)

00:10 We'll use this formula in our exercise

00:14 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:

$\frac{\left(x\times y\right)^{27}}{\left(x\times y\right)^{20}}=$

Step-by-step solution

To solve the problem, we need to simplify the expression $\frac{\left(x\times y\right)^{27}}{\left(x\times y\right)^{20}}$ .

We will apply the Power of a Quotient Rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ .

Let's denote $a = x \times y$ , and our expression becomes $\frac{a^{27}}{a^{20}}$ . According to the rule:

$m = 27$
$n = 20$

We subtract the exponents: $m - n = 27 - 20 = 7$ .

Thus, $a^{m-n} = a^7 = (x \times y)^7$ .

Therefore, the expression simplifies to $\left(x \times y\right)^7$ .

The solution to the question is:

$\left(x\times y\right)^7$

Final Answer

$\left(x\times y\right)^7$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing same bases, subtract the exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ so 27 - 20 = 7
Check: Verify that $(xy)^7 \times (xy)^{20} = (xy)^{27}$ ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents during division
Don't add exponents like 27 + 20 = 47 or multiply like 27 × 20 = 540! Division of same bases requires subtraction of exponents. Always use the rule: subtract the bottom exponent from the top exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it as canceling out repeated multiplication! When you have $\frac{a^{27}}{a^{20}}$ , you're canceling 20 copies of 'a' from both top and bottom, leaving $a^{27-20} = a^7$ .

What if the bottom exponent is larger than the top?

You still subtract! For example, $\frac{x^3}{x^5} = x^{3-5} = x^{-2}$ . The negative exponent means you have a fraction: $\frac{1}{x^2}$ .

Does (x×y) count as one base or two?

It counts as one single base! Since both terms have the same base $(x \times y)$ , you can apply the division rule directly.

How do I remember which operation to use?

Use this memory trick: Division = Subtraction, Multiplication = Addition. When you see a fraction bar (÷), think subtract!

Can I simplify the expression inside the parentheses first?

No need to! The beauty of exponent rules is that they work with any base, whether it's a single variable, a product like (xy), or even a complex expression.

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