How to simplify (x^5)^-8x^-4

Question

Going into College Prep Algebra 2 Junior Year...

 

How do you simplify (x⁵)^-8x⁴ ?

Answer 1

Chantal,

 

Combining multiple exponents works like this:

 

For a number raised to a power then raised to another power:  (x^a)^b  = x^(a·b)

 

and x^a · x^b = x^(a+b)

 

Example:  (x²)³ = x⁶  and x²·x³ = x⁵

 

so for you example (x⁵)^-8(x⁴) = (x^-40)(x⁴) = x^-36 

 

where we employed operation #1 (a·b) first, followed by operation # 2 (a+b), to get the solution.

Answer 2

Use rules of exponents. First of all   

 

   (x⁵)^-8 = x ^-40

 

Then if your next term is x⁴  you will have x^-40 x⁴ = x^-36  (you are adding the exponents)

 

If you have x ^-4  then your final answer is x ^-44.

Answer 3

(x⁵)^-8x⁴

 

To simplify, you'll need to apply the following rules of exponents:

1. (x^a)^b = x^ab
2. x^ax^b = x^a+b and x^a/x^b = x^a-b
3. x^-a = 1/x^a

 

Use rule 1 first:

 

(x⁵)^-8x⁴

 

x^-40x⁴

 

Now rule 2:

 

x^-40x⁴ = x^-40+4 = x^-36

 

Now rule 3:

 

x^-36 = 1/x³⁶

Answer 4

      Monomials ARE a pain, but manageable if you know the rules. When a monomial is raised to a power, you can rewrite it using a single power which is the product of the original 2 powers. By the way, you simplify in the order of the order of operations, so exponents are done before multiplication. So, x to the 5th raised to the -8th is the same as x^-40. When you multiply monomials with the same base , you add the exponents, so x^-40 *x^4 = x^-36. Your teacher probably wants you to express your answer using positive exponents. Raising a number to a negative exponent is equivalent to raing its reciprocal to the positive exponent.

For example (2/5)^-3 = (5/2)^3 which is 5^3/2^3 =125/8.

x^-36 = (1/x)^36 = 1^36/x^36 = 1/x^36.

 

Confusing, but I hope this helps.

Answer 5

Chantal:

 

(x⁵)⁸x⁴ = [(x⁵) (x⁵).... 8 times] x⁴  = x^{(5+5+... 8 times)} x⁴ = x ^{40 + 4} = x⁴⁴.

 

I showed you above why the simplification of  (x^m)ⁿ is (x^m)ⁿ = x^mn.

 

Dattaprabhakar (Dr .G.)