Simplify the Expression: 8^4 × 8 × 8^-1 Using Exponent Laws

Q: Why is 8 written as 8¹ in the solution?

Any number without a written exponent has an invisible exponent of 1. So $ 8 = 8^1 $. This makes it easier to apply the exponent rule!

Q: How do I handle negative exponents when adding?

Treat negative exponents like negative numbers in addition. So 4 + 1 + (-1) = 4 + 1 - 1 = 4. The negative sign stays with the exponent.

Q: What if I multiply the exponents instead of adding them?

That's a common mistake! Multiplying exponents is only used for power of a power like $ (8^2)^3 $. For multiplying same bases, always add the exponents.

Q: What's the difference between 8^(-1) and -8^1?

$ 8^{-1} $ means one divided by 8 (which equals 1/8), while $ -8^1 $ means negative 8. The negative sign's position matters!

Exponent Laws with Negative Exponents

Insert the corresponding expression:

$8^4\times8\times8^{-1}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this problem step by step.

00:13 Remember, any number to the power of one is just the number itself.

00:18 So, we'll use this rule and raise our number to the power of one.

00:22 When we multiply powers with the same base, like A, we add their exponents.

00:27 This means A to the power of N plus M. Let's try this now.

00:32 In our problem, we're also adding a negative exponent.

00:37 Remember, a positive times a negative gives a negative, so we'll subtract.

00:43 All right, let's see the solution!

00:49 And that's how we solve it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$8^4\times8\times8^{-1}=$

Step-by-step solution

To solve this problem, we will apply the multiplication of powers rule which states that when multiplying powers with the same base, we add their exponents.

Let's begin by analyzing the given expression: $8^4 \times 8 \times 8^{-1}$ .

Each term has the base 8, allowing us to use the exponent rule directly:

First, recognize the exponents for each term: the first term $8^4$ has an exponent of 4, the second term $8$ is equivalent to $8^1$ , and the third term $8^{-1}$ has an exponent of -1.
Then, apply the formula by adding the exponents: $4 + 1 - 1$ .

The resulting expression for the exponent is $8^{4+1-1}$ .

Therefore, the corresponding expression to the original product is $8^{4+1-1}$ .

Final Answer

$8^{4+1-1}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: Convert $8$ to $8^1$ , then add: 4 + 1 + (-1)
Check: Final exponent 4 + 1 - 1 = 4, so result is $8^4$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply exponents like 4 × 1 × (-1) = -4! This gives $8^{-4}$ instead of the correct answer. When multiplying powers with the same base, always add the exponents: 4 + 1 + (-1) = 4.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why is 8 written as 8¹ in the solution?

Any number without a written exponent has an invisible exponent of 1. So $8 = 8^1$ . This makes it easier to apply the exponent rule!

How do I handle negative exponents when adding?

Treat negative exponents like negative numbers in addition. So 4 + 1 + (-1) = 4 + 1 - 1 = 4. The negative sign stays with the exponent.

What if I multiply the exponents instead of adding them?

That's a common mistake! Multiplying exponents is only used for power of a power like $(8^2)^3$ . For multiplying same bases, always add the exponents.

Can I simplify this further after getting 8^(4+1-1)?

Yes! Calculate the exponent: 4 + 1 - 1 = 4, so the final simplified form is $8^4$ . You could even calculate $8^4 = 4096$ if needed.

What's the difference between 8^(-1) and -8^1?

$8^{-1}$ means one divided by 8 (which equals 1/8), while $-8^1$ means negative 8. The negative sign's position matters!

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