Simplify the Complex Fraction: Dividing Powers and Managing Negative Exponents

Video Solution

Solution Steps

00:00 Select the appropriate sign

00:03 When multiplying powers with equal bases

00:08 The power of the result equals the sum of the powers

00:12 We'll apply this formula to our exercise, let's add up the powers

00:23 We'll do the same thing on the right side of the exercise

00:31 Let's calculate the powers

00:53 In order to eliminate a negative power

00:56 We'll invert the numerator and denominator and the power will become positive

00:59 We'll apply this formula to our exercise, and transfer from the numerator to the denominator

01:23 Let's identify where the power is larger

01:30 Only this power changes because the rest of the exercise is the same

01:38 The larger power is in the denominator, therefore it's smaller

01:42 This is the solution

Step-by-Step Solution

Let's systematically simplify both expressions and then compare them:

Simplifying the First Expression:

$\frac{7^2 \cdot 7^{-8}}{7^3 \cdot (-7)^4}$

Apply Product of Powers Rule to the numerator: $7^2 \cdot 7^{-8} = 7^{2 + (-8)} = 7^{-6}$ .
Use Power of a Power Rule in the denominator for $(-7)^4 = (-1)^4 \cdot 7^4 = 7^4$ because $(-1)^4 = 1$ .
Simplify the denominator: $7^3 \cdot 7^4 = 7^{3+4} = 7^7$ .
Apply Quotient of Powers Rule: $\frac{7^{-6}}{7^7} = 7^{-6-7} = 7^{-13}$ .

Simplifying the Second Expression:

$\frac{7^2 \cdot 7^{-9}}{7^3 \cdot (-7)^4}$

Apply Product of Powers Rule to the numerator: $7^2 \cdot 7^{-9} = 7^{2 + (-9)} = 7^{-7}$ .
The denominator is the same as before: $7^3 \cdot 7^4 = 7^7$ .
Apply Quotient of Powers Rule: $\frac{7^{-7}}{7^7} = 7^{-7-7} = 7^{-14}$ .

Comparison:

Therefore, the first expression is greater than the second expression. The correct choice is: $>$ .