Solve (2×4×8)^(a+3): Understanding Compound Power Expressions

Q: Why can't I just multiply 2×4×8 = 64 first?

While $ 2\times4\times8 = 64 $ is correct, writing $ 64^{a+3} $ loses the factor structure needed for algebraic manipulation. You need to keep the individual bases separate for further operations.

Q: What's the difference between the correct answer and option A?

Option A shows $ 2^a4^38^{a+3} $ with different exponents on each base. The correct answer has $ 2^{a+3}4^{a+3}8^{a+3} $ - the same exponent (a+3) on all three bases.

Q: Does the order of the factors matter?

No! Multiplication is commutative, so $ 2^{a+3}\cdot4^{a+3}\cdot8^{a+3} $ equals $ 4^{a+3}\cdot2^{a+3}\cdot8^{a+3} $ or any other arrangement of these factors.

Q: Can I simplify this expression further?

Yes! Since $ 4 = 2^2 $ and $ 8 = 2^3 $, you could rewrite everything in terms of base 2: $ 2^{a+3}\cdot(2^2)^{a+3}\cdot(2^3)^{a+3} = 2^{6a+18} $

Q: What if the exponent was just a number, like 3?

The same rule applies! $ (2\cdot4\cdot8)^3 = 2^3\cdot4^3\cdot8^3 = 8\cdot64\cdot512 $. You can then calculate the final numerical answer if needed.

Power Distribution with Multiple Base Factors

$(2\cdot4\cdot8)^{a+3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's solve this problem together.

00:13 When you have a power on a product, raise each term to that power.

00:22 We'll use this rule in our exercise.

00:27 Raise each factor to the power, one step at a time.

00:32 And that's how you find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(2\cdot4\cdot8)^{a+3}=$

Step-by-step solution

Let's begin by using the distributing exponents rule (An exponent outside of a parentheses needs to be distributed across all the numbers and variables within the parentheses)

$(x\cdot y)^n=x^n\cdot y^n$ We first apply this rule to the given problem:

$(2\cdot4\cdot8)^{a+3}= 2^{a+3}4^{a+3}8^{a+3}$ When then we apply the power to each of the terms of the product inside the parentheses separately and maintain the multiplication.

The correct answer is option d.

Final Answer

$2^{a+3}4^{a+3}8^{a+3}$

Key Points to Remember

Essential concepts to master this topic

Distribution Rule: Exponent applies to each factor in the product separately
Technique: $(2\cdot4\cdot8)^{a+3} = 2^{a+3}\cdot4^{a+3}\cdot8^{a+3}$
Check: Each base factor must have the same exponent (a+3) ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to the product result instead of individual factors
Don't calculate (2×4×8) = 64 first, then write 64^(a+3)! This completely changes the algebraic structure and makes further simplification impossible. Always distribute the exponent to each individual factor: 2^(a+3) × 4^(a+3) × 8^(a+3).

Practice Quiz

Test your knowledge with interactive questions

Choose the expression(s) that corresponds to the following:

$\left(5\times6\right)^9=$

FAQ

Everything you need to know about this question

Why can't I just multiply 2×4×8 = 64 first?

While $2\times4\times8 = 64$ is correct, writing $64^{a+3}$ loses the factor structure needed for algebraic manipulation. You need to keep the individual bases separate for further operations.

What's the difference between the correct answer and option A?

Option A shows $2^a4^38^{a+3}$ with different exponents on each base. The correct answer has $2^{a+3}4^{a+3}8^{a+3}$ - the same exponent (a+3) on all three bases.

Does the order of the factors matter?

No! Multiplication is commutative, so $2^{a+3}\cdot4^{a+3}\cdot8^{a+3}$ equals $4^{a+3}\cdot2^{a+3}\cdot8^{a+3}$ or any other arrangement of these factors.

Can I simplify this expression further?

Yes! Since $4 = 2^2$ and $8 = 2^3$ , you could rewrite everything in terms of base 2: $2^{a+3}\cdot(2^2)^{a+3}\cdot(2^3)^{a+3} = 2^{6a+18}$

What if the exponent was just a number, like 3?

The same rule applies! $(2\cdot4\cdot8)^3 = 2^3\cdot4^3\cdot8^3 = 8\cdot64\cdot512$ . You can then calculate the final numerical answer if needed.

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Solve (2×4×8)^(a+3): Understanding Compound Power Expressions

Power Distribution with Multiple Base Factors

❤️ 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 multiply 2×4×8 = 64 first?

What's the difference between the correct answer and option A?

Does the order of the factors matter?

Can I simplify this expression further?

What if the exponent was just a number, like 3?

🌟 Unlock Your Math Potential

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