Math Problems: Exponents And Logarithms Explained

Oct 14, 2025 by Blender 50 views

Hey guys! Let's break down these math problems step by step. We're going to cover exponents and logarithms, making sure everything is super clear. Get ready to sharpen those pencils, and let's dive right in!

2.2. (0-1) Understanding Exponents: Writing 5³ Correctly

Okay, so the question asks us how to represent 5³ (5 cubed). This is all about understanding what exponents mean. When you see 5³, it means you're multiplying 5 by itself three times:

5 * 5 * 5

Let's evaluate each option to see which one matches:

A. 5.55: This is just a decimal number and has nothing to do with exponents. It's clearly not the right answer.
B. 253/5: To figure this out, we need to do the division. 253 divided by 5 is 50.6. Again, this doesn't represent 5³.
C. 125/5: Let's divide 125 by 5. 125 / 5 = 25. Wait a minute! 125 looks familiar. That's because 5 * 5 * 5 = 125, meaning that 5³ = 125, so to express this in the required form where it has a denominator of 5, the correct option would be 125/1 to directly represent 5³. However, since 125/5 isn't directly equal to 5³, there seems to be some ambiguity, as 5³ = 125. I will assume here the question intends 5³ to be exactly equivalent to 125/1 and we can consider the operation 125/5 as a manipulation of the exponentiation's result, making it difficult to decide if this is intended to represent the original question accurately.

So, after careful review of the options, it seems there's an issue with how the question is posed or the available answers. The direct numerical calculation of 5³ gives us 125, which is not directly present as an exact option in the format asked (something / 5), making the interpretation somewhat complex.

Therefore, let's reinterpret the question:

If the student is supposed to find an equivalent expression to the result of 5³, where that result is then divided by 5, then we are asking: what is 5³/5?

Given that 5³ = 125, then 5³/5 = 125/5 = 25. However, that does not directly answer the question either, because it solves for 5³/5, not 5³.

Conclusion:

Based on typical interpretations of exponents, 5³ equals 125. The multiple-choice options provided don't directly represent 5³ in an equivalent fractional form with a denominator, leading to potential confusion. If we strictly evaluate 5³, the result is 125, but none of the given options clearly match this direct interpretation.

Final Answer:

Considering the possible ambiguity, it's crucial to seek clarification on the intended interpretation of the question. None of the options directly and clearly represent 5³ in a fractional form.

2.3. (0-1) Evaluating Numerical Expressions: Solving 27/(-3)²

Next up, we have to figure out what 27/(-3)² equals. Remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We'll start with the exponent:

(-3)²: This means -3 multiplied by itself: (-3) * (-3) = 9. A negative number times a negative number becomes positive.

Now we can rewrite the expression:

27 / 9: This is a simple division problem. 27 divided by 9 equals 3.

Let's look at the options:

A. -3: Nope, we got a positive 3.
B. -18: Definitely not.
C. 1: Wrong again.

Wait a second! None of the options match our answer. Our calculation clearly shows that 27/(-3)² = 3, but 3 isn't one of the choices. There seems to be a mistake in the question or the provided answers.

Therefore, let's double-check our work.

Recalculating (-3)²: (-3) * (-3) = 9. This is correct.
Recalculating 27 / 9: 27 / 9 = 3. This is also correct.

Conclusion:

The expression 27/(-3)² evaluates to 3. The provided options (A. -3, B. -18, C. 1) do not include the correct answer. It's possible there was a typo in the original question or the answer choices.

Final Answer:

The correct answer, based on the expression provided, is 3, which is not listed in the given options. There may be an error in the question or answer choices.

2.4. (0-1) Simplifying Logarithmic Expressions

Alright, this one looks a bit more complex. We need to simplify this expression:

(log₂(100) - 2log₂2) / (log₂50 + log₂2)

Let's break it down step by step, using logarithm properties. Remember these key rules:

logₐ(b) + logₐ(c) = logₐ(bc)*
logₐ(b) - logₐ(c) = logₐ(b/c)
nlogₐ(b) = logₐ(bⁿ)*

Step 1: Simplify the numerator

We have log₂(100) - 2log₂2. Let's use the power rule to rewrite 2log₂2:

2log₂2 = log₂(2²)= log₂(4)

Now the numerator becomes:

log₂(100) - log₂(4)

Using the quotient rule:

log₂(100) - log₂(4) = log₂(100/4) = log₂(25)

Step 2: Simplify the denominator

We have log₂50 + log₂2. Using the product rule:

log₂50 + log₂2 = log₂(50 * 2) = log₂(100)

Step 3: Rewrite the entire expression

Now we have:

log₂(25) / log₂(100)

Step 4: Use the change of base formula (optional, but helpful)

We can rewrite this as:

log₂(25) / log₂(100) = log₁₀(25) / log₁₀(100) <- Here we have converted from log base 2 to log base 10.

Notice that 100 = 4 * 25, therefore:

log₂100 = log₂(4*25) = log₂4 + log₂25

So, let's go back to our expression

log₂(25) / log₂(100) = log₂(25) / (log₂4 + log₂25)

Now, let x = log₂25, our expression can be rewritten as:

x/(log₂4 + x)

Since log₂4 = 2 (because 2² = 4), we have

x/(2 + x)

Now let's analyze the given options:

A. 3 + log₂4
B. 3 + log₂4

Since both options are the same, let's evaluate if our simplified expression, x/(2+x), can be manipulated to look like that. 3 + log₂4 = 3 + 2 = 5. We are looking to see if log₂(25)/log₂(100) is always equal to 5, which is very unlikely.

Conclusion: The question seems to have an error, as it does not lead to either of the proposed solutions. log₂(25) / log₂(100) is not immediately simplifiable to a numerical value or the form offered in the answers.

Final Answer:

Based on the simplification and analysis, the given options (A. 3 + log₂4 and B. 3 + log₂4) do not directly result from simplifying the original logarithmic expression. The problem may contain an error or require more advanced logarithmic manipulations not immediately apparent.