Compare Complex Expressions: ((x×y)⁵)¹² vs ((x×y)¹⁰)²)³

Q: Why do I multiply the exponents instead of adding them?

The power of a power rule says $ (a^m)^n = a^{m \times n} $. This is different from multiplying powers where you add exponents. Think of it as: if you have 5 groups of something, and you take that 12 times, you get 5 × 12 = 60 total!

Q: How do I handle multiple layers of parentheses like in the second expression?

Work from the inside out! First simplify $ ((xy)^{10})^2 = (xy)^{20} $, then raise that to the 3rd power: $ ((xy)^{20})^3 = (xy)^{60} $.

Q: What if the expressions had different bases like x²y and xy²?

Then you cannot directly compare them using exponent rules alone! You'd need specific values for x and y. But since both expressions here have the same base $ (x \times y) $, we can compare the exponents.

Q: Can I solve this without fully simplifying both sides?

It's much safer to simplify both sides completely first. This way you can clearly see if the exponents are the same (equal), or which one is larger. Comparing complex nested expressions directly is very error-prone!

Q: What does the × symbol mean between x and y?

The × symbol means multiplication, just like writing xy. So $ x \times y $ and $ xy $ mean exactly the same thing - we're treating xy as one combined base.

Step-by-step video solution

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Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the compatible sign:

$>,<,=$

$\left(\left(x\times y\right)^5\right)^{12}\Box\left(\left(\left(x\times y\right)^{10}\right)^2\right)^3$

2

Step-by-step solution

To solve the problem, we need to simplify and compare the following two expressions:

$\left(\left(x \times y\right)^5\right)^{12} \quad \text{and} \quad \left(\left(\left(x \times y\right)^{10}\right)^2\right)^3$

Let's simplify each expression:

The first expression $\left(\left(x \times y\right)^5\right)^{12}$ .
Using the power of a power rule, $\left(a^m\right)^n = a^{m \cdot n}$ , we have:

$\left(\left(x \times y\right)^5\right)^{12} = \left(x \times y\right)^{5 \cdot 12} = \left(x \times y\right)^{60}$

The second expression $\left(\left(\left(x \times y\right)^{10}\right)^2\right)^3$ .
Similarly, apply the power of a power rule twice:

$\left(\left(\left(x \times y\right)^{10}\right)^2\right)^3 = \left(\left(x \times y\right)^{10 \cdot 2}\right)^3$

$= \left(x \times y\right)^{20 \cdot 3} = \left(x \times y\right)^{60}$

After simplification, both expressions become $\left(x \times y\right)^{60}$ .

Therefore, the relationship between the two expressions is:

$\left(x \times y\right)^{60} = \left(x \times y\right)^{60}$

=

Thus, the correct choice from the provided options is:

: =

I am confident that the solution is correct, as both expressions simplify to the same value.

3

Final Answer

=

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to another power, multiply the exponents
Technique: For $(a^m)^n$ , the result is $a^{m \times n}$
Check: Both expressions simplify to $(x \times y)^{60}$ , so they are equal ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add exponents like 5 + 12 = 17 when working with nested powers! This gives completely wrong results like $(xy)^{17}$ instead of $(xy)^{60}$ . Always multiply exponents when raising a power to another power: $(a^m)^n = a^{m \times n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

+

The power of a power rule says $(a^m)^n = a^{m \times n}$ . This is different from multiplying powers where you add exponents. Think of it as: if you have 5 groups of something, and you take that 12 times, you get 5 × 12 = 60 total!

How do I handle multiple layers of parentheses like in the second expression?

+

Work from the inside out! First simplify $((xy)^{10})^2 = (xy)^{20}$ , then raise that to the 3rd power: $((xy)^{20})^3 = (xy)^{60}$ .

What if the expressions had different bases like x²y and xy²?

+

Then you cannot directly compare them using exponent rules alone! You'd need specific values for x and y. But since both expressions here have the same base $(x \times y)$ , we can compare the exponents.

Can I solve this without fully simplifying both sides?

+

It's much safer to simplify both sides completely first. This way you can clearly see if the exponents are the same (equal), or which one is larger. Comparing complex nested expressions directly is very error-prone!

What does the × symbol mean between x and y?

+

The × symbol means multiplication, just like writing xy. So $x \times y$ and $xy$ mean exactly the same thing - we're treating xy as one combined base.

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Compare Complex Expressions: ((x×y)⁵)¹² vs ((x×y)¹⁰)²)³

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