Solve: 4^a × 8^a × 2^a - Multiplying Powers with Same Exponent

Q: Why can I multiply the bases when the exponents are the same?

This follows the reverse of the power of a product rule! Since $ (abc)^n = a^n \times b^n \times c^n $, we can work backwards: $ a^n \times b^n \times c^n = (abc)^n $.

Q: What if the exponents were different, like $ 4^a \times 8^b \times 2^c $ ?

Then you cannot combine them this way! The power rule only works when all exponents are identical. Different exponents mean the expression stays as separate terms.

Q: Do I need to calculate 4 × 8 × 2 = 64 in my final answer?

It depends on the question! If asked for the simplified form, write $ 64^a $. If asked for the equivalent expression, $ (4 \times 8 \times 2)^a $ is perfect.

Q: Can I use this rule with more than three terms?

Absolutely! The rule works with any number of terms: $ a^n \times b^n \times c^n \times d^n = (abcd)^n $. Just make sure all exponents are the same.

Q: What's the difference between this and $ a^4 \times a^8 \times a^2 $ ?

Great question! When the bases are the same but exponents differ, you add the exponents: $ a^4 \times a^8 \times a^2 = a^{14} $. Here, bases differ but exponents are the same, so you multiply the bases.

Power Rule with Multiple Bases

Insert the corresponding expression:

$4^a\times8^a\times2^a=$

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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 According to the laws of exponents, a product raised to a power (N)

00:07 Equals a product where each factor is raised to the same power (N)

00:10 We will apply this formula to our exercise

00:13 Note that each factor is raised to the same power (N)

00:22 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:

$4^a\times8^a\times2^a=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the common exponent in all terms.
Step 2: Apply the power of a product rule to combine the terms.

Now, let's work through each step:
Step 1: We have the expression $4^a \times 8^a \times 2^a$ , where each base is raised to the same power, $a$ .
Step 2: Using the power of a product rule, we can combine the terms into a single expression: $(4 \times 8 \times 2)^a$ .
Therefore, in terms of the choice list, the corresponding expression is $\left(4\times8\times2\right)^a$ , which is choice 2.

Final Answer

$\left(4\times8\times2\right)^a$

Key Points to Remember

Essential concepts to master this topic

Rule: Same exponents allow bases to be multiplied first
Technique: $4^a \times 8^a \times 2^a = (4 \times 8 \times 2)^a$
Check: Calculate base product first: 4 × 8 × 2 = 64, so $64^a$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying bases
Don't write $4^a \times 8^a \times 2^a = 14^{3a}$ by adding bases and multiplying exponents! This confuses different exponent rules. Always multiply the bases when exponents are the same: $(4 \times 8 \times 2)^a$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I multiply the bases when the exponents are the same?

This follows the reverse of the power of a product rule! Since $(abc)^n = a^n \times b^n \times c^n$ , we can work backwards: $a^n \times b^n \times c^n = (abc)^n$ .

What if the exponents were different, like $4^a \times 8^b \times 2^c$ ?

Then you cannot combine them this way! The power rule only works when all exponents are identical. Different exponents mean the expression stays as separate terms.

Do I need to calculate 4 × 8 × 2 = 64 in my final answer?

It depends on the question! If asked for the simplified form, write $64^a$ . If asked for the equivalent expression, $(4 \times 8 \times 2)^a$ is perfect.

Can I use this rule with more than three terms?

Absolutely! The rule works with any number of terms: $a^n \times b^n \times c^n \times d^n = (abcd)^n$ . Just make sure all exponents are the same.

What's the difference between this and $a^4 \times a^8 \times a^2$ ?

Great question! When the bases are the same but exponents differ, you add the exponents: $a^4 \times a^8 \times a^2 = a^{14}$ . Here, bases differ but exponents are the same, so you multiply the bases.

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Solve: 4^a × 8^a × 2^a - Multiplying Powers with Same Exponent

Power Rule with Multiple Bases

❤️ 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 I multiply the bases when the exponents are the same?

What if the exponents were different, like $4^a \times 8^b \times 2^c$ ?

Do I need to calculate 4 × 8 × 2 = 64 in my final answer?

Can I use this rule with more than three terms?

What's the difference between this and $a^4 \times a^8 \times a^2$ ?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Simplify the Product: 3^x × 7^x × 5^x Using Exponent Rules

Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

Simplify the Expression: 6^t × 9^t Using Laws of Exponents

Simplify the Expression: 5^y × 3^y Using Exponent Properties

Solve: 4^a × 8^a × 2^a - Multiplying Powers with Same Exponent

Power Rule with Multiple Bases

❤️ 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 I multiply the bases when the exponents are the same?

What if the exponents were different, like 4a×8b×2c 4^a \times 8^b \times 2^c 4a×8b×2c?

Do I need to calculate 4 × 8 × 2 = 64 in my final answer?

Can I use this rule with more than three terms?

What's the difference between this and a4×a8×a2 a^4 \times a^8 \times a^2 a4×a8×a2?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Simplify the Product: 3^x × 7^x × 5^x Using Exponent Rules

Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

Simplify the Expression: 6^t × 9^t Using Laws of Exponents

Simplify the Expression: 5^y × 3^y Using Exponent Properties

What if the exponents were different, like $4^a \times 8^b \times 2^c$ ?

What's the difference between this and $a^4 \times a^8 \times a^2$ ?