Simplify 8^(-10) × 8^(-5) × 8^9: Combining Like Base Exponents

Q: Why do I add the exponents instead of multiplying them?

The multiplication rule for exponents says $ a^m \times a^n = a^{m+n} $. When bases are the same, you add exponents because you're combining repeated multiplication!

Q: What's the difference between $ 8^{-6} $ and $ \frac{1}{8^6} $ ?

They're exactly the same! The negative exponent rule says $ a^{-n} = \frac{1}{a^n} $, so $ 8^{-6} = \frac{1}{8^6} $.

Q: Can I simplify $ \frac{1}{8^6} $ further?

You could calculate $ 8^6 = 262,144 $ to get $ \frac{1}{262,144} $, but usually $ \frac{1}{8^6} $ is the preferred form as it's cleaner.

Q: What if I had different bases like $ 2^3 \times 4^2 $ ?

You can't directly add exponents with different bases. First convert to the same base: $ 4 = 2^2 $, so $ 4^2 = (2^2)^2 = 2^4 $, then add exponents.

Exponent Rules with Negative Powers

Insert the corresponding expression:

$8^{-10}\times8^{-5}\times8^9=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, the multiplication of powers with the same base (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:11 We will apply this formula to our exercise

00:15 We'll maintain the base and add the exponents together

00:19 Given that we are multiplying a negative factor, we must be careful with the parentheses

00:36 A positive x A negative always equals a negative, therefore we subtract as follows

00:40 Let's calculate

00:48 According to the laws of exponents, any number with a negative exponent (-N)

00:51 equals its reciprocal raised to the opposite exponent (N)

00:54 We will apply this formula to our exercise

00:57 We'll convert to the reciprocal number and raise it to the opposite exponent

01:01 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:

$8^{-10}\times8^{-5}\times8^9=$

Step-by-step solution

To solve this problem, we need to simplify the expression $8^{-10} \times 8^{-5} \times 8^9$ using exponent rules.

Step 1: Apply the multiplication rule for exponents. This rule states that when multiplying expressions with the same base, you add their exponents. Thus, we calculate:
$8^{-10} \times 8^{-5} \times 8^9 = 8^{-10 + (-5) + 9}$ .
Step 2: Simplify the exponents:
$-10 + (-5) + 9 = -10 - 5 + 9 = -6$ .
Step 3: The expression simplifies to:
$8^{-6}$ .
Step 4: Convert the negative exponent into a positive one by using the rule for negative exponents, where $a^{-n} = \frac{1}{a^n}$ :
$8^{-6} = \frac{1}{8^6}$ .

Therefore, the simplified expression is $\frac{1}{8^6}$ .

The corresponding expression is:

$\frac{1}{8^6}$

Final Answer

$\frac{1}{8^6}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add all exponents together
Technique: Calculate -10 + (-5) + 9 = -6 step by step
Check: Convert $8^{-6}$ to $\frac{1}{8^6}$ using negative exponent rule ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply -10 × (-5) × 9 = 450! This treats exponents like regular multiplication and gives a completely wrong answer. Always add exponents when multiplying expressions with the same base: -10 + (-5) + 9 = -6.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The multiplication rule for exponents says $a^m \times a^n = a^{m+n}$ . When bases are the same, you add exponents because you're combining repeated multiplication!

How do I handle negative exponents in the calculation?

Treat negative exponents like negative numbers when adding: $-10 + (-5) = -15$ , then $-15 + 9 = -6$ . Be careful with your signs!

What's the difference between $8^{-6}$ and $\frac{1}{8^6}$ ?

They're exactly the same! The negative exponent rule says $a^{-n} = \frac{1}{a^n}$ , so $8^{-6} = \frac{1}{8^6}$ .

Can I simplify $\frac{1}{8^6}$ further?

You could calculate $8^6 = 262,144$ to get $\frac{1}{262,144}$ , but usually $\frac{1}{8^6}$ is the preferred form as it's cleaner.

What if I had different bases like $2^3 \times 4^2$ ?

You can't directly add exponents with different bases. First convert to the same base: $4 = 2^2$ , so $4^2 = (2^2)^2 = 2^4$ , then add exponents.

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Simplify 8^(-10) × 8^(-5) × 8^9: Combining Like Base Exponents

Exponent Rules with Negative Powers

❤️ 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 do I add the exponents instead of multiplying them?

How do I handle negative exponents in the calculation?

What's the difference between $8^{-6}$ and $\frac{1}{8^6}$ ?

Can I simplify $\frac{1}{8^6}$ further?

What if I had different bases like $2^3 \times 4^2$ ?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Solve: 4^(-6) × 4 Using Negative Exponent Rules

Multiply Powers with Base 5: Solving 5^(-8) × 5^6

Multiply and Simplify: 9^-1 × 9^-2 × 9^-3 Using Exponent Rules

Simplify the Expression: 11^-2 × 11^-5 × 11^-4 Using Exponent Laws

Simplify 8^(-10) × 8^(-5) × 8^9: Combining Like Base Exponents

Exponent Rules with Negative Powers

❤️ 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 do I add the exponents instead of multiplying them?

How do I handle negative exponents in the calculation?

What's the difference between 8−6 8^{-6} 8−6 and 186 \frac{1}{8^6} 861​?

Can I simplify 186 \frac{1}{8^6} 861​ further?

What if I had different bases like 23×42 2^3 \times 4^2 23×42?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Solve: 4^(-6) × 4 Using Negative Exponent Rules

Multiply Powers with Base 5: Solving 5^(-8) × 5^6

Multiply and Simplify: 9^-1 × 9^-2 × 9^-3 Using Exponent Rules

Simplify the Expression: 11^-2 × 11^-5 × 11^-4 Using Exponent Laws

What's the difference between $8^{-6}$ and $\frac{1}{8^6}$ ?

Can I simplify $\frac{1}{8^6}$ further?

What if I had different bases like $2^3 \times 4^2$ ?