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

Q: Why does 2⁰ equal 1 instead of 0?

The zero exponent rule states that any non-zero base raised to the power of 0 equals 1. Think of it as dividing by itself: $ 2^3 ÷ 2^3 = 2^{3-3} = 2^0 = 1 $!

Q: How do I know when to convert bases?

Look for numbers that can be written as powers of the same base. Here, 9 = 3², so converting $ 9^2 $ to $ 3^4 $ lets you combine with $ 3^{-4} $!

Q: What happens when I have negative exponents?

Negative exponents mean "flip to denominator": $ 3^{-4} = \frac{1}{3^4} $. When multiplying, they combine with positive exponents: $ 3^{-4} \times 3^4 = 3^0 = 1 $.

Q: Why does the final answer have 3⁸ in the denominator?

When you move $ 3^{-4} $ to the denominator, it becomes $ 3^4 $. Combined with the existing $ 3^4 $ from $ 9^2 $, you get $ 3^4 \times 3^4 = 3^8 $!

Q: Can I simplify this fraction further?

No, $ \frac{1}{5^4 \cdot 3^8} $ is in simplest form because 5 and 3 share no common factors. The numerator is 1, so there's nothing more to reduce.

Exponent Rules with Base Conversions

Solve the following problem:

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following problem:

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

Step-by-step 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 $\frac{1}{5^4 \cdot 3^8}$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Zero Exponent Rule: Any base to power of 0 equals 1
Base Conversion: Convert $9^2 = (3^2)^2 = 3^4$ to match bases
Check Powers: Verify $3^{-4} \times 3^4 = 3^0 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to convert 9 to base 3
Don't leave $9^2$ as is = you miss combining like bases! This prevents simplification and leads to more complex answers. Always convert to matching bases like $9 = 3^2$ first.

Practice Quiz

Test your knowledge with interactive questions

Which of the following is equivalent to $$ 100^0 $$ ?

FAQ

Everything you need to know about this question

Why does 2⁰ equal 1 instead of 0?

The zero exponent rule states that any non-zero base raised to the power of 0 equals 1. Think of it as dividing by itself: $2^3 ÷ 2^3 = 2^{3-3} = 2^0 = 1$ !

How do I know when to convert bases?

Look for numbers that can be written as powers of the same base. Here, 9 = 3², so converting $9^2$ to $3^4$ lets you combine with $3^{-4}$ !

What happens when I have negative exponents?

Negative exponents mean "flip to denominator": $3^{-4} = \frac{1}{3^4}$ . When multiplying, they combine with positive exponents: $3^{-4} \times 3^4 = 3^0 = 1$ .

Why does the final answer have 3⁸ in the denominator?

When you move $3^{-4}$ to the denominator, it becomes $3^4$ . Combined with the existing $3^4$ from $9^2$ , you get $3^4 \times 3^4 = 3^8$ !

Can I simplify this fraction further?

No, $\frac{1}{5^4 \cdot 3^8}$ is in simplest form because 5 and 3 share no common factors. The numerator is 1, so there's nothing more to reduce.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

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

Exponent Rules with Base Conversions

❤️ 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 does 2⁰ equal 1 instead of 0?

How do I know when to convert bases?

What happens when I have negative exponents?

Why does the final answer have 3⁸ in the denominator?

Can I simplify this fraction further?

🌟 Unlock Your Math Potential

More Questions

Exponents Rules

Simplify the Expression: (9² × 3⁻⁴) ÷ 6³ Step by Step

Solve 7⁴ × 8³ × (1/7)⁴: Power and Reciprocal Multiplication

Examine the Power of Inversion: Simplify 300^-4 x (1/300)^-4

Simplify 7^x × 7^(-x): Multiplying Exponential Expressions

Zero Exponent Rule

Evaluate 4^5 - 4^6 × (1/4): Powers and Operations Problem

Solve (9·3)÷8^0: Zero Exponent Division Problem

Solve (300 × 5/3 × 2/7)⁰: Understanding the Zero Power Rule

Solve for the Exponent: (7/4)ˣ = 1

Applying Combined Exponents Rules

Solve: 3^(-3) × 19^35 × 19^(-32) ÷ 19^3 | Exponent Simplification

Solving Radical Expression: Simplify ⁶√b¹²×(1/b)²×a

Evaluate 112^0: Understanding the Zero Exponent Rule

Simplify This Complex Expression: (13/2)^0 ⋅ (2/13)^(-2) ⋅ (13/2)^(-5)