Calculate (13×4)^6: Evaluating a Power of Products Expression

Q: Why can't I just multiply 13 × 4 = 52 first?

While calculating $ 52^6 $ gives the same numerical result, the question asks for the equivalent algebraic expression. The power of a product rule shows $ (13 \times 4)^6 = 13^6 \times 4^6 $.

Q: What is the power of a product rule exactly?

The rule states: $ (a \times b)^n = a^n \times b^n $. When you have a product inside parentheses raised to a power, distribute that exponent to each factor separately.

Q: Does this work with more than two factors?

Yes! For example: $ (2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4 $. The exponent distributes to every single factor in the product.

Q: How do I remember which factors get the exponent?

Think: "What's inside the parentheses?" Everything inside gets the exponent. In $ (13 \times 4)^6 $, both 13 and 4 are inside, so both get raised to the 6th power.

Power of Products with Exponent Distribution

Choose the expression that corresponds to the following:

$\left(13\times4\right)^6=$

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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 In order to expand parentheses containing a multiplication operation with an outside exponent

00:06 Raise each factor to the power

00:09 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

Choose the expression that corresponds to the following:

$\left(13\times4\right)^6=$

Step-by-step solution

The expression in question is $\left(13\times4\right)^6$ . This expression involves raising a product to a power and requires the application of the power of a product exponent rule, which states:

When you have a product raised to an exponent, you can distribute the exponent to each factor in the product separately.

Mathematically, any numbers $a$ and $b$ and a positive integer $n$ can be written as $(a \times b)^n = a^n \times b^n$ .

Applying this rule to our expression, $\left(13 \times 4\right)^6$ becomes $13^6 \times 4^6$ .

Therefore, the expression $\left(13 \times 4\right)^6$ simplifies to $13^6 \times 4^6$ .

Final Answer

$13^6\times4^6$

Key Points to Remember

Essential concepts to master this topic

Rule: Distribute the exponent to each factor in the product
Technique: $(13 \times 4)^6 = 13^6 \times 4^6$ using exponent distribution
Check: Both expressions equal the same value when calculated ✓

Common Mistakes

Avoid these frequent errors

Computing the product first then raising to the power
Don't calculate 13 × 4 = 52 first to get $52^6$ ! This gives the correct numerical answer but doesn't match the required algebraic form. Always distribute the exponent to each factor separately: $(13 \times 4)^6 = 13^6 \times 4^6$ .

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 13 × 4 = 52 first?

While calculating $52^6$ gives the same numerical result, the question asks for the equivalent algebraic expression. The power of a product rule shows $(13 \times 4)^6 = 13^6 \times 4^6$ .

What is the power of a product rule exactly?

The rule states: $(a \times b)^n = a^n \times b^n$ . When you have a product inside parentheses raised to a power, distribute that exponent to each factor separately.

Does this work with more than two factors?

Yes! For example: $(2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4$ . The exponent distributes to every single factor in the product.

Why is 13^6 × 4 incorrect?

This only applies the exponent 6 to the factor 13, leaving 4 unchanged. The rule requires both factors to be raised to the 6th power: $13^6 \times 4^6$ .

How do I remember which factors get the exponent?

Think: "What's inside the parentheses?" Everything inside gets the exponent. In $(13 \times 4)^6$ , both 13 and 4 are inside, so both get raised to the 6th power.

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