Calculate (10×3)⁴: Fourth Power of a Product Expression

Q: Why are all three answers correct?

All three expressions are mathematically equivalent! $ (10 \times 3)^4 $ can be written as $ 10^4 \times 3^4 $ using the power rule, as $ 3^4 \times 10^4 $ by commutative property, or as $ 30^4 $ by simplifying first.

Q: When do I use the power of a product rule?

Use this rule whenever you see parentheses with multiplication raised to a power. The rule $ (a \times b)^n = a^n \times b^n $ lets you distribute the exponent to each factor inside the parentheses.

Q: Is it better to simplify inside parentheses first or apply the rule?

Both methods work perfectly! You can calculate $ 10 \times 3 = 30 $ first to get $ 30^4 $, or apply the power rule to get $ 10^4 \times 3^4 $. Choose whichever feels easier!

Q: Does order matter in multiplication?

No! The commutative property means $ 10^4 \times 3^4 $ equals $ 3^4 \times 10^4 $. You can multiply factors in any order and get the same result.

Q: How can I check if expressions are equivalent?

Calculate the numerical value of each expression! If they all give the same final number, they're equivalent. For example, $ 30^4 = 810000 $ and $ 10^4 \times 3^4 = 10000 \times 81 = 810000 $.

Power Rules with Product Expressions

Choose the expression that corresponds to the following:

$\left(10\times3\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this expression together.

00:11 We'll try two different methods. First, we'll solve what's inside the parentheses.

00:18 This is the first method.

00:20 Now, let's explore the second simplification method.

00:24 To expand parentheses with an exponent over multiplication,

00:29 Raise each factor inside to the power.

00:32 We'll apply this formula in our exercise now.

00:38 Remember, in multiplication, the order of factors doesn't matter.

00:43 So, both expressions are equal.

00:46 We'll use the formula again and switch the factors accordingly.

00:53 Again, let's use the formula to simplify the exponent on multiplication.

00:59 And there you go! That's how we solve this problem.

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:

$\left(10\times3\right)^4=$

Step-by-step solution

To solve this problem, we'll apply the power of a product rule to the expression $(10 \times 3)^4$ .

Step 1: Identify the expression.
The given expression is $(10 \times 3)^4$ .
Step 2: Apply the power of a product law.
According to the rule, $(a \times b)^n = a^n \times b^n$ , our expression becomes:
$(10 \times 3)^4 = 10^4 \times 3^4$ .
Step 3: Evaluate the choices:
- First choice: $3^4 \times 10^4$
Rearranging terms, this is equivalent to $10^4 \times 3^4$ . Therefore, it matches our transformed expression.
- Second choice: $30^4$
Since $30 = 10 \times 3$ , $(10 \times 3)^4 = 30^4$ . This simplifies to the same expression.
- Third choice: $10^4 \times 3^4$
Therefore, the solution is that all answers are correct.

Final Answer

All of the above

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(a \times b)^n = a^n \times b^n$
Technique: $(10 \times 3)^4 = 10^4 \times 3^4$ and also equals $30^4$
Check: Verify all equivalent forms give same value: $10^4 \times 3^4 = 3^4 \times 10^4 = 30^4$ ✓

Common Mistakes

Avoid these frequent errors

Thinking only one form is correct
Don't choose just one answer when multiple expressions are mathematically equivalent = missing correct solutions! Students often forget that multiplication is commutative and that simplifying inside parentheses first also works. Always recognize that $(a \times b)^n$ , $a^n \times b^n$ , and $(ab)^n$ are all equivalent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are all three answers correct?

All three expressions are mathematically equivalent! $(10 \times 3)^4$ can be written as $10^4 \times 3^4$ using the power rule, as $3^4 \times 10^4$ by commutative property, or as $30^4$ by simplifying first.

When do I use the power of a product rule?

Use this rule whenever you see parentheses with multiplication raised to a power. The rule $(a \times b)^n = a^n \times b^n$ lets you distribute the exponent to each factor inside the parentheses.

Is it better to simplify inside parentheses first or apply the rule?

Both methods work perfectly! You can calculate $10 \times 3 = 30$ first to get $30^4$ , or apply the power rule to get $10^4 \times 3^4$ . Choose whichever feels easier!

Does order matter in multiplication?

No! The commutative property means $10^4 \times 3^4$ equals $3^4 \times 10^4$ . You can multiply factors in any order and get the same result.

How can I check if expressions are equivalent?

Calculate the numerical value of each expression! If they all give the same final number, they're equivalent. For example, $30^4 = 810000$ and $10^4 \times 3^4 = 10000 \times 81 = 810000$ .

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