Question

This is from Khan academy I have to attach a PNG if you can help me solve it! Thank you!

Answer 1

Step-1) Simplifying the expression

Given expression:
`64^(m) /{4^(2m) }`

We can rewrite the expression (in the numerator and the denominator) as the product of multiple fours. Then, we can apply the exponent rule to simplify the expression to its simplest form. The simplest form will be the required simplified expression (solution to the provided expression).

• 
  `\implies (4^(3)) ^(m) /{4^(2m) } \\`
• 
  `\implies 4^(3m) /{4^(2m) }`

We can apply the following exponent rule to simplify the expression:

`\boxed{\text{Exponent rule:} \ 4^(m) /4^(n) = 4^(m - n)}`

The exponent rule states that the "base" must be the same when subtracting exponents. If we divide a term with same bases, we can reduce work time by subtracting the exponent to simplify the expression.

• 
  `\implies 4^(3m) /{4^(2m) }`
• 
  `\implies 4^(3m - 2m) = 4^(m)`

Step-2) Equivalent or Non-equivalent?

Now, let us look at all the options to verify which term matches our simplified term, and which expressions do not match our simplified term.

First option:

Given term:
`2^(2m) \\`

Can be re-written as:

• 
  `=2^(2m) \\`
• 
  `= [2^((2))]^(m)`

Simplifying the expression inside the long brackets:

• 
  `\\= [4]^(m)`
• 
  `\\= 4^(m) \ ( \text{matches})`

Therefore, the first option is equivalent to our simplified term.

Second option:

Given term:
`16^(0.5m)`

Can be re-written as:

• 
  `= 16^(0.5m)`
• 
  `\\= (4^(2)) ^(0.5m)`

Exponent Rule: (xᵃ)ᵇ = xᵃᵇ

• 
  `= (4^(1)) ^m`
• 
  `= (4^(1m))`
• 
  `= 4^m`

Therefore, the second option is equivalent to our simplified term.

Third option:

Given term:
`4^(m)`

This term already matches our simplified term.

Therefore, the third option is equivalent to our simplified term.

Step-3) Conclude/verify your answer

We can conclude that all the options provided are equivalent to the given expression. We proved it by applying exponent rules and formulas.

Answer 2

Explanation:

Question

64^m

--------

4^2m

Solution

(64)^m = (4^3)^m = 4^3m

4^3m/4^2m = 4^(3m - 2m) = 4^m

Answer

2^2m = (2^2)^m = 4^m Equivalent

16^0.5m = (16^0.4) ^m = 4^m Equivalent

4^m Equivalent

All three of these are equivalent. The catch is in splitting the powers apart. In this top one (2^2m) you move the brackets so that the right bracket is after the two which gives (2^2)^m = 4m

You do the same thing with (16^0.5)^m. 16^0.5 = 4 So the answer is 4^m

The last one is 4^m which is the answer you got from the division.