Simplify the Expression: k²·t⁴·k⁶·t² Using Exponent Rules

Q: Why can't I just add all the exponents together?

The rule $ a^m \cdot a^n = a^{m+n} $ only works for identical bases. Since k and t are different variables, you must treat them separately. Think of it like counting apples and oranges - you can't combine them into one total!

Q: Do I need to rearrange the terms before solving?

It's not required, but it's highly recommended! Grouping like terms together (all k's, then all t's) makes it much easier to see which exponents to add. This prevents confusion and mistakes.

Q: What if the bases were numbers instead of variables?

The same rule applies! For example: $ 2^3 \cdot 2^5 = 2^{3+5} = 2^8 = 256 $. Whether bases are numbers or variables, add the exponents when multiplying identical bases.

Q: Can this expression be simplified further?

No, $ k^8 \cdot t^6 $ is fully simplified! Since k and t are different variables, they cannot be combined any further. This is the final answer.

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

Remember: Multiplication with same bases: ADD exponentsPower of a power: MULTIPLY exponentsIn this problem, we're multiplying terms with same bases, so we add!

Exponent Laws with Multiple Variables

$k^2\cdot t^4\cdot k^6\cdot t^2=$

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Understand the problem

$k^2\cdot t^4\cdot k^6\cdot t^2=$

Step-by-step solution

Using the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$ It is important to note that this law is only valid for terms with identical bases,

We notice that in the problem there are two types of terms. First, for the sake of order, we will use the substitution property to rearrange the expression so that the two terms with the same base are grouped together. The, we will proceed to solve:

$k^2t^4k^6t^2=k^2k^6t^4t^2$ Next, we apply the power property to each different type of term separately,

$k^2k^6t^4t^2=k^{2+6}t^{4+2}=k^8t^6$ We apply the property separately - for the terms whose bases are $k$ and for the terms whose bases are $t$ We add the powers in the exponent when we multiply all the terms with the same base.

The correct answer then is option b.

Final Answer

$k^8\cdot t^6$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add exponents: $a^m \cdot a^n = a^{m+n}$
Technique: Group same bases together: $k^2 \cdot k^6 = k^{2+6} = k^8$
Check: Verify by counting each variable separately: k appears 2+6=8 times, t appears 4+2=6 times ✓

Common Mistakes

Avoid these frequent errors

Adding exponents across different bases
Don't add all exponents together like 2+4+6+2=14! This gives $kt^{14}$ which is completely wrong. The exponent law only applies to terms with identical bases. Always group k terms separately from t terms, then add exponents within each group.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just add all the exponents together?

The rule $a^m \cdot a^n = a^{m+n}$ only works for identical bases. Since k and t are different variables, you must treat them separately. Think of it like counting apples and oranges - you can't combine them into one total!

Do I need to rearrange the terms before solving?

It's not required, but it's highly recommended! Grouping like terms together (all k's, then all t's) makes it much easier to see which exponents to add. This prevents confusion and mistakes.

What if the bases were numbers instead of variables?

The same rule applies! For example: $2^3 \cdot 2^5 = 2^{3+5} = 2^8 = 256$ . Whether bases are numbers or variables, add the exponents when multiplying identical bases.

Can this expression be simplified further?

No, $k^8 \cdot t^6$ is fully simplified! Since k and t are different variables, they cannot be combined any further. This is the final answer.

How do I remember when to add vs multiply exponents?

Remember:

Multiplication with same bases: ADD exponents
Power of a power: MULTIPLY exponents

In this problem, we're multiplying terms with same bases, so we add!

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Simplify the Expression: k²·t⁴·k⁶·t² Using Exponent Rules

Exponent Laws with Multiple Variables

❤️ 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't I just add all the exponents together?

Do I need to rearrange the terms before solving?

What if the bases were numbers instead of variables?

Can this expression be simplified further?

How do I remember when to add vs multiply exponents?

🌟 Unlock Your Math Potential

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