Algebraic Expressions: Step-by-Step Solutions

Let's break down these algebraic expressions step by step, guys! We'll make sure everything is super clear and easy to follow. Let's dive in!

1) 100⁵ ÷ 1000²

When solving exponential expressions like this, the key is to get everything to the same base. It makes the math way easier. Here's how we tackle it:

• Express everything in terms of a common base: Notice that both 100 and 1000 are powers of 10. We can rewrite them as 100 = 10² and 1000 = 10³.
• Substitute the values: Replace 100 and 1000 in the original expression: (10²)⁵ ÷ (10³)².
• Apply the power of a power rule: Remember that (aᵇ)ᶜ = aᵇᶜ. So, we get 10²⁵ ÷ 10³² which simplifies to 10¹⁰ ÷ 10⁶.
• Apply the quotient of powers rule: When dividing powers with the same base, you subtract the exponents: aᵇ ÷ aᶜ = aᵇ⁻ᶜ. Therefore, 10¹⁰ ÷ 10⁶ = 10¹⁰⁻⁶ = 10⁴.
• Calculate the final value: 10⁴ is simply 10,000.

So, 100⁵ ÷ 1000² = 10,000. Isn't that neat how it all simplifies down? Understanding the rules of exponents is super important here.

Let's think about why expressing everything in terms of a common base is so crucial. It allows us to directly compare and manipulate the exponents, which are the heart of these expressions. Without a common base, we'd be stuck trying to compare apples and oranges. Also, remember the power of a power rule. It's one of the most frequently used rules when simplifying expressions with exponents.

In summary, breaking down the problem into smaller, manageable steps by using rules of exponents and common bases ensures accuracy and makes the process less intimidating. Always look for opportunities to simplify before diving into calculations!

2) (3¹⁰ × (3³)⁵) / ((3⁵)⁴ × 3)

Okay, this one looks a bit more complex, but don't sweat it! We're going to use the same exponent rules to simplify it step by step. Here we go:

• Simplify the numerator: First, let's tackle the numerator: 3¹⁰ × (3³)⁵. Using the power of a power rule, (3³)⁵ becomes 3¹⁵. So the numerator is now 3¹⁰ × 3¹⁵.
• Apply the product of powers rule to the numerator: When multiplying powers with the same base, you add the exponents: aᵇ × aᶜ = aᵇ⁺ᶜ. Therefore, 3¹⁰ × 3¹⁵ = 3²⁵.
• Simplify the denominator: Now, let's simplify the denominator: (3⁵)⁴ × 3. Again, using the power of a power rule, (3⁵)⁴ becomes 3²⁰. So the denominator is now 3²⁰ × 3.
• Apply the product of powers rule to the denominator: Remember that 3 is the same as 3¹. So, 3²⁰ × 3¹ = 3²¹.
• Simplify the entire expression: Now we have 3²⁵ / 3²¹. Using the quotient of powers rule, we subtract the exponents: 3²⁵⁻²¹ = 3⁴.
• Calculate the final value: 3⁴ = 3 × 3 × 3 × 3 = 81.

Therefore, (3¹⁰ × (3³)⁵) / ((3⁵)⁴ × 3) = 81. See? Not so scary when you break it down!

This type of problem emphasizes the importance of order of operations and methodical application of rules. Dealing with exponents often involves multiple steps, and rushing can easily lead to mistakes. By tackling each part of the expression independently, we minimize the chance of error. Also, it's a good idea to double-check each step to ensure accuracy, especially when dealing with multiple exponents.

In summary, remember to simplify each part separately, apply the exponent rules carefully, and double-check your work. Staying organized and patient is the key to success!

3) (4³ × 16²) / 2¹²

Alright, let's simplify this expression. Again, the trick is to express everything in terms of a common base. In this case, we can use 2 as the base, since both 4 and 16 are powers of 2.

• Express everything in terms of base 2: We know that 4 = 2² and 16 = 2⁴. So, we can rewrite the expression as ((2²)³ × (2⁴)²) / 2¹².
• Apply the power of a power rule: (2²)³ becomes 2⁶ and (2⁴)² becomes 2⁸. The expression now looks like (2⁶ × 2⁸) / 2¹².
• Apply the product of powers rule: Multiply the terms in the numerator: 2⁶ × 2⁸ = 2¹⁴.
• Simplify the entire expression: Now we have 2¹⁴ / 2¹². Using the quotient of powers rule, we subtract the exponents: 2¹⁴⁻¹² = 2².
• Calculate the final value: 2² = 4.

So, (4³ × 16²) / 2¹² = 4. Awesome!

The importance of recognizing common bases cannot be overstated. By converting all numbers to the same base, we unlocked the simplicity hidden within the expression. This technique is particularly useful when dealing with various powers and roots. Moreover, understanding how to manipulate exponents not only simplifies complex arithmetic but also lays a strong foundation for more advanced algebraic concepts.

In summary, by finding the common base and simplifying the expression by applying power rules, we made what initially looked complex, manageable and simple.

4) 45¹⁰ / (5⁸ × 3¹⁹)

This problem might seem intimidating at first, but let's break it down. The trick here is to recognize that 45 can be expressed as a product of 5 and 3 (45 = 5 × 3²). This will allow us to simplify the expression using exponent rules.

• Express 45 in terms of its prime factors: Since 45 = 5 × 3², we can rewrite 45¹⁰ as (5 × 3²)¹⁰.
• Apply the power of a product rule: (5 × 3²)¹⁰ = 5¹⁰ × (3²)¹⁰ = 5¹⁰ × 3²⁰.
• Rewrite the original expression: Now we have (5¹⁰ × 3²⁰) / (5⁸ × 3¹⁹).
• Apply the quotient of powers rule: Divide the powers with the same base by subtracting the exponents: 5¹⁰ / 5⁸ = 5¹⁰⁻⁸ = 5² and 3²⁰ / 3¹⁹ = 3²⁰⁻¹⁹ = 3¹.
• Simplify the expression: We are left with 5² × 3¹ = 25 × 3.
• Calculate the final value: 25 × 3 = 75.

Therefore, 45¹⁰ / (5⁸ × 3¹⁹) = 75. Great job!

Problems like this teach us the value of prime factorization in simplifying expressions. Identifying the fundamental building blocks of numbers often reveals hidden relationships and allows us to apply exponent rules more effectively. Moreover, this problem underscores the importance of pattern recognition and creative problem-solving in algebra. By recognizing that 45 could be broken down into simpler factors, we transformed a seemingly difficult problem into a straightforward calculation.

In summary, prime factorization makes a difference in how we see problems. It allows us to break down composite numbers and apply the rules of exponents more creatively.