Solve (8×2)^x: Complete the Power Expression

Q: Why can't I just multiply the numbers inside first?

You could simplify $ (8 \times 2)^x = 16^x $, but the question asks for the expanded form using the power of product rule. This shows you understand how exponents distribute over multiplication.

Q: What's the difference between this and adding exponents?

Adding exponents applies when you have same bases: $ a^m \times a^n = a^{m+n} $. Here we have different bases (8 and 2) with the same exponent, so we use the power of product rule instead.

Q: Does this work with more than two numbers?

Yes! The rule extends to any number of factors: $ (a \times b \times c)^n = a^n \times b^n \times c^n $. Each factor gets raised to the same power.

Q: How do I remember when to use this rule?

Look for parentheses around multiplication with an exponent outside. The key pattern is $ (\text{factor} \times \text{factor})^{\text{power}} $ - this tells you to distribute the power to each factor.

Q: What if the exponent was a number instead of x?

The same rule applies! For example, $ (8 \times 2)^3 = 8^3 \times 2^3 = 512 \times 8 = 4096 $. Whether the exponent is a variable or number, distribute it to each factor.

Power Rules with Product Expressions

Insert the corresponding expression:

$\left(8\times2\right)^x=$

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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 open parentheses with a multiplication operation and an outside exponent

00:07 We raise each factor to the power

00:10 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

Insert the corresponding expression:

$\left(8\times2\right)^x=$

Step-by-step solution

Let's solve the problem by applying the power of a product rule.

Step 1: Recognize the product within the exponent. The original expression is $(8 \times 2)^x$ .
Step 2: Apply the power of a product formula, which is $(a \times b)^n = a^n \times b^n$ .

By applying this formula, we get:

$(8 \times 2)^x = 8^x \times 2^x$

Therefore, the expanded form of $(8 \times 2)^x$ is $8^x \times 2^x$ .

Thus, the solution to this problem is $8^x \times 2^x$ .

Final Answer

$8^x\times2^x$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply $(a \times b)^n = a^n \times b^n$ to distribute exponents
Technique: $(8 \times 2)^x$ becomes $8^x \times 2^x$ by power of product rule
Check: Verify by testing with x = 2: $(16)^2 = 256$ and $8^2 \times 2^2 = 64 \times 4 = 256$ ✓

Common Mistakes

Avoid these frequent errors

Distributing exponent to only one factor or multiplying exponent by factors
Don't write $(8 \times 2)^x = 8 \times 2 \times x$ or $8^x \times 2$ = wrong result! The exponent multiplies with factors incorrectly or only applies to one part. Always distribute the exponent to each factor: $(a \times b)^n = a^n \times b^n$ .

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 the numbers inside first?

You could simplify $(8 \times 2)^x = 16^x$ , but the question asks for the expanded form using the power of product rule. This shows you understand how exponents distribute over multiplication.

What's the difference between this and adding exponents?

Adding exponents applies when you have same bases: $a^m \times a^n = a^{m+n}$ . Here we have different bases (8 and 2) with the same exponent, so we use the power of product rule instead.

Does this work with more than two numbers?

Yes! The rule extends to any number of factors: $(a \times b \times c)^n = a^n \times b^n \times c^n$ . Each factor gets raised to the same power.

How do I remember when to use this rule?

Look for parentheses around multiplication with an exponent outside. The key pattern is $(\text{factor} \times \text{factor})^{\text{power}}$ - this tells you to distribute the power to each factor.

What if the exponent was a number instead of x?

The same rule applies! For example, $(8 \times 2)^3 = 8^3 \times 2^3 = 512 \times 8 = 4096$ . Whether the exponent is a variable or number, distribute it to each factor.

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