Calculate (2×4)^10: Evaluating Powers with Parentheses

Q: Why does the exponent apply to both numbers inside the parentheses?

The parentheses mean we're raising the entire product to the 10th power. Think of it as $ (2 \times 4) \times (2 \times 4) \times ... $ ten times, which gives us $ 2^{10} \times 4^{10} $.

Q: Can I simplify $ (2 \times 4)^{10} $ to $ 8^{10} $ first?

Yes! You could calculate $ 2 \times 4 = 8 $ first to get $ 8^{10} $. But the question asks for the equivalent expression, so $ 2^{10} \times 4^{10} $ is the correct form.

Q: What's wrong with $ 2^{10} \times 4 $ as an answer?

This only applies the exponent to the first number! The power of a product rule requires the exponent to apply to both factors: $ 2^{10} \times 4^{10} $.

Q: How do I remember this rule?

Think: "Exponents distribute to everyone in parentheses!" Just like distributing candy to everyone at a party, the exponent goes to each factor inside the parentheses.

Q: Does this work with more than two numbers?

Absolutely! For example, $ (2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4 $. The exponent applies to every factor inside the parentheses.

Power of Product with Exponent Rules

Choose the expression that corresponds to the following:

$\left(2\times4\right)^{10}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:11 To tackle a multiplication inside parentheses with an exponent outside,

00:16 Raise each factor inside to the power of the exponent.

00:24 Now, let's apply this rule to our exercise.

00:33 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(2\times4\right)^{10}=$

Step-by-step solution

To solve the question, we need to apply the power of a product exponent rule. The formula states that for any real numbers $a$ and $b$ , and any integer $n$ :

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

Looking at our expression, we can see that:

$a = 2$
$b = 4$
$n = 10$

Now, if we apply the formula:

$(2 \times 4)^{10} = 2^{10} \times 4^{10}$

Therefore, the expression $(2 \times 4)^{10}$ is equivalent to $2^{10} \times 4^{10}$ .

Final Answer

$2^{10}\times4^{10}$

Key Points to Remember

Essential concepts to master this topic

Rule: Power of a product applies exponent to each factor
Technique: $(a \times b)^n = a^n \times b^n$ distributes the exponent
Check: Both factors must have same exponent: $2^{10} \times 4^{10}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of applying to both factors
Don't write $(2 \times 4)^{10}$ as $2^5 \times 4^2$ or split the exponent = wrong answer! This confuses addition with distribution. Always apply the full exponent to each factor: $2^{10} \times 4^{10}$ .

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

Why does the exponent apply to both numbers inside the parentheses?

The parentheses mean we're raising the entire product to the 10th power. Think of it as $(2 \times 4) \times (2 \times 4) \times ...$ ten times, which gives us $2^{10} \times 4^{10}$ .

Can I simplify $(2 \times 4)^{10}$ to $8^{10}$ first?

Yes! You could calculate $2 \times 4 = 8$ first to get $8^{10}$ . But the question asks for the equivalent expression, so $2^{10} \times 4^{10}$ is the correct form.

What's wrong with $2^{10} \times 4$ as an answer?

This only applies the exponent to the first number! The power of a product rule requires the exponent to apply to both factors: $2^{10} \times 4^{10}$ .

How do I remember this rule?

Think: "Exponents distribute to everyone in parentheses!" Just like distributing candy to everyone at a party, the exponent goes to each factor inside the parentheses.

Does this work with more than two numbers?

Absolutely! For example, $(2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4$ . The exponent applies to every factor inside the parentheses.

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Calculate (2×4)^10: Evaluating Powers with Parentheses

Power of Product with Exponent Rules

❤️ 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 does the exponent apply to both numbers inside the parentheses?

Can I simplify $(2 \times 4)^{10}$ to $8^{10}$ first?

What's wrong with $2^{10} \times 4$ as an answer?

How do I remember this rule?

Does this work with more than two numbers?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Evaluate (5×6)^9: Solving a Multiple-Step Exponent Problem

Evaluate (20×5)^7: Solving a Complex Exponent Expression

Calculate the Cube Power: (25×4)³ Step-by-Step Solution

Evaluate (2×3)² : Order of Operations with Square Numbers

Calculate (2×4)^10: Evaluating Powers with Parentheses

Power of Product with Exponent Rules

❤️ 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 does the exponent apply to both numbers inside the parentheses?

Can I simplify (2×4)10 (2 \times 4)^{10} (2×4)10 to 810 8^{10} 810 first?

What's wrong with 210×4 2^{10} \times 4 210×4 as an answer?

How do I remember this rule?

Does this work with more than two numbers?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Evaluate (5×6)^9: Solving a Multiple-Step Exponent Problem

Evaluate (20×5)^7: Solving a Complex Exponent Expression

Calculate the Cube Power: (25×4)³ Step-by-Step Solution

Evaluate (2×3)² : Order of Operations with Square Numbers

Can I simplify $(2 \times 4)^{10}$ to $8^{10}$ first?

What's wrong with $2^{10} \times 4$ as an answer?