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

Insert the corresponding expression:

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

Video Solution

Solution Steps

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 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:

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