Multiply Powers: Simplifying t^6 × 4^6 Expression

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

Multiplying exponents (getting 36) applies to different rules! The rule $ (a^m)^n = a^{mn} $ is for powers raised to powers, not multiplication of same powers.

Q: What if the exponents were different?

If exponents are different, like $ t^6 \times 4^3 $, you cannot combine them using this rule. The bases must have the exact same exponent to use this property.

Q: Does order matter when combining bases?

No! Whether you write $ (t \times 4)^6 $ or $ (4 \times t)^6 $, both are correct because multiplication is commutative.

Q: How do I remember this exponent rule?

Think of it as "same power, combine bases". When you see identical exponents, you can group the bases together under one exponent: $ a^n \times b^n = (ab)^n $.

Q: Can this work with more than two terms?

Yes! For example: $ 2^3 \times x^3 \times y^3 = (2xy)^3 $. As long as all terms have the same exponent, you can combine any number of bases.

Exponent Rules with Same Powers

Insert the corresponding expression:

$t^6\times4^6=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 A product where each factor is raised to that factor's power (N)

00:07 Can be converted to parentheses of the entire product raised to the factor (N)

00:11 We will apply this formula to our exercise

00:16 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$t^6\times4^6=$

Step-by-step solution

To solve this problem, we'll utilize the property of exponents that states:

$(a^n \times b^n) = (a \times b)^n$

This property allows us to combine terms that are each raised to the same power. In this case, we have:

$t^6$ and $4^6$ both raised to the power of 6.

By applying the property, we can rewrite the expression:

$t^6 \times 4^6 = (t \times 4)^6$ .

This simplifies the original expression by combining the bases under a single exponent.

Therefore, the expression equivalent to $t^6 \times 4^6$ is $\left(t \times 4\right)^6$ .

Final Answer

$\left(t\times4\right)^6$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases have same exponent, $a^n \times b^n = (a \times b)^n$
Technique: Combine $t^6 \times 4^6$ into $(t \times 4)^6$ using property
Check: Expand both forms to verify they equal the same result ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't add the exponents to get $(t \times 4)^{12}$ = completely wrong answer! This confuses the multiplication rule with the power-of-power rule. Always combine bases when exponents are the same, keeping the original exponent.

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 multiply the exponents together?

Multiplying exponents (getting 36) applies to different rules! The rule $(a^m)^n = a^{mn}$ is for powers raised to powers, not multiplication of same powers.

What if the exponents were different?

If exponents are different, like $t^6 \times 4^3$ , you cannot combine them using this rule. The bases must have the exact same exponent to use this property.

Does order matter when combining bases?

No! Whether you write $(t \times 4)^6$ or $(4 \times t)^6$ , both are correct because multiplication is commutative.

How do I remember this exponent rule?

Think of it as "same power, combine bases". When you see identical exponents, you can group the bases together under one exponent: $a^n \times b^n = (ab)^n$ .

Can this work with more than two terms?

Yes! For example: $2^3 \times x^3 \times y^3 = (2xy)^3$ . As long as all terms have the same exponent, you can combine any number of bases.

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