Simplify the Expression: (2⁰×3⁻⁴)/(5⁴×9²) with Mixed Exponents

Question

Solve the following problem:

$\frac{2^0\cdot3^{-4}}{5^4\cdot9^2}=\text{?}$

00:00 Simplify the following problem

00:03 According to the exponent laws, any number when raised to the power of 0 equals 1

00:06 As long as the number is not 0

00:10 Apply this formula to our exercise

00:25 With any fraction/number with a negative exponent

00:28 The numerator and denominator can be flipped in order to obtain a positive exponent

00:35 Apply this formula to our exercise

00:52 Break down 9 to 3 squared

00:58 When there's a power of a power, the combined exponent is the product of the exponents

01:04 We will once again apply this formula to our exercise, we'll then proceed to multiply the exponents

01:21 When multiplying powers with equal bases

01:24 The exponent of the result equals the sum of the exponents

01:29 We will apply this formula in our exercise and add the exponents together

01:40 This is the solution

In order to solve the given problem, we will follow these steps:

Step 1: Simplify $2^0$ . According to the zero exponent rule, $2^0 = 1$ .
Step 2: Simplify $3^{-4}$ . Using the negative exponent rule, $3^{-4} = \frac{1}{3^4}$ .
Step 3: Simplify $9^2$ . Recognize that $9 = 3^2$ , thus $9^2 = (3^2)^2 = 3^{4}$ .
Step 4: Substitute the simplified terms back into the expression:

$\frac{2^0 \cdot 3^{-4}}{5^4 \cdot 9^2} = \frac{1 \cdot \frac{1}{3^4}}{5^4 \cdot 3^{4}}$

Step 5: Simplify by combining like bases: since $3^{-4}$ in the numerator can be combined with $3^4$ in the denominator, you have:

$= \frac{1}{5^4 \cdot 3^{4+4}} = \frac{1}{5^4 \cdot 3^8}$

Therefore, the simplified expression is $\frac{1}{5^4 \cdot 3^8}$ .

$\frac{1}{5^4\cdot3^8}$