4. find all the powers of 4 in the range 4 through 1,000.
what is the difference between 4x and ( x^4 )? evaluate both of these expressions when ( x = 2 ).
lesson 5: exponents

Question

Accepted Answer

### First Sub - Question: Find all the powers of 4 in the range 4 through 1,000.
# Explanation:
## Step 1: Recall the formula for powers of a number
The $n$ - th power of 4 is given by $4^{n}$, where $n$ is a positive integer. We start with $n = 1$ and keep increasing $n$ until $4^{n}>1000$.
For $n = 1$: $4^{1}=4$
## Step 2: Calculate $4^{2}$
$4^{2}=4\times4 = 16$
## Step 3: Calculate $4^{3}$
$4^{3}=4\times4\times4=64$
## Step 4: Calculate $4^{4}$
$4^{4}=4\times4\times4\times4 = 256$
## Step 5: Calculate $4^{5}$
$4^{5}=4\times4\times4\times4\times4=1024$
Since $1024>1000$, we stop here.
# Answer: The powers of 4 in the range 4 through 1000 are $4^{1} = 4$, $4^{2}=16$, $4^{3}=64$, $4^{4}=256$

### Second Sub - Question: What is the difference between $4x$ and $x^{4}$? Evaluate both of these expressions when $x = 2$.
#### Part 1: Difference between $4x$ and $x^{4}$
# Brief Explanations:
$4x$ is a linear expression (a first - degree polynomial) where the variable $x$ is multiplied by 4. $x^{4}$ is a quartic expression (a fourth - degree polynomial) where the variable $x$ is raised to the power of 4. The difference between them is $x^{4}-4x$ (or $4x - x^{4}$ depending on the order, but generally, we can express the difference as $x^{4}-4x$).
#### Part 2: Evaluate when $x = 2$
# Explanation:
## Step 1: Evaluate $4x$ when $x = 2$
Substitute $x = 2$ into $4x$. Using the formula $4x$, we get $4\times2=8$
## Step 2: Evaluate $x^{4}$ when $x = 2$
Substitute $x = 2$ into $x^{4}$. Using the formula $x^{4}$, we get $2^{4}=2\times2\times2\times2 = 16$
## Step 3: Find the difference between the two results
The difference between $x^{4}$ and $4x$ when $x = 2$ is $16 - 8=8$ (if we take $x^{4}-4x$) or $8 - 16=-8$ (if we take $4x - x^{4}$). But usually, when we talk about the difference between two expressions $A$ and $B$, we can also just present the two values and then the difference.
# Answer:
- The expression $4x$ is a linear term ($4$ times $x$) and $x^{4}$ is a quartic term ($x$ to the power of $4$). The difference is $x^{4}-4x$ (or $4x - x^{4}$).
- When $x = 2$, $4x=8$ and $x^{4}=16$. The difference between $x^{4}$ and $4x$ is $8$ (or $- 8$ if we consider $4x - x^{4}$).

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QUESTION IMAGE

Question

Explanation:

First Sub - Question: Find all the powers of 4 in the range 4 through 1,000.

Step 1: Recall the formula for powers of a number

Step 2: Calculate \(4^{2}\)

Step 3: Calculate \(4^{3}\)

Step 4: Calculate \(4^{4}\)

Step 5: Calculate \(4^{5}\)

Part 2: Evaluate when \(x = 2\)

Step 1: Evaluate \(4x\) when \(x = 2\)

Step 2: Evaluate \(x^{4}\) when \(x = 2\)

Step 3: Find the difference between the two results

Answer:

Second Sub - Question: What is the difference between \(4x\) and \(x^{4}\)? Evaluate both of these expressions when \(x = 2\).

Part 1: Difference between \(4x\) and \(x^{4}\)