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

Reduce the following equation:

$11^{-2}\times11^{-5}\times11^{-4}=$

Video Solution

Solution Steps

00:00 Simplify the following problem

00:03 According to the laws of exponents, the multiplication of powers with equal bases (A)

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

00:10 We'll apply this formula to our exercise

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

00:19 Note that we're adding a negative factor, be careful with the parentheses

00:31 A positive x A negative always equals a negative, therefore we subtract as follows

00:42 According to the laws of exponents, any number with a negative exponent (-N)

00:45 equals the reciprocal number raised to the opposite exponent (N)

00:48 We'll apply this formula to our exercise

00:51 We'll convert to the reciprocal number and raise it to the opposite exponent

00:55 This is the solution

Step-by-Step Solution

To solve the expression $11^{-2} \times 11^{-5} \times 11^{-4}$ , we apply the rules for multiplying numbers with the same base:

Step 1: Use the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ .
Step 2: Add the exponents: $-2 + -5 + -4$ .
Step 3: Perform the calculation: $-2 - 5 - 4 = -11$ .
Step 4: Write the expression with the combined exponent: $11^{-11}$ .
Step 5: Express $11^{-11}$ as a positive power using the property of negative exponents: $a^{-n} = \frac{1}{a^n}$ .

Therefore, $11^{-11} = \frac{1}{11^{11}}$ .

The final answer is $\frac{1}{11^{11}}$ .