Find whole numbers a and b that make the equation true.

Question

(3^a*8²)⁵=3¹⁵*8^b

Answer 1

On the left side of equation, we multiply the terms in parenthesis by itself 5 times.  When we do this, we are multiplying terms with common bases.   Add the exponents of common bases.

 

(3^a * 8²)(3^a * 8²)(3^a * 8²)(3^a * 8²)(3^a* 8²) = 3¹⁵ * 8^b

 

3^5a * 8¹⁰ = 3¹⁵ * 8^b

 

 

We want the left and right side of equation to be equal to each other. 

 

3^5a = 3¹⁵                                           8¹⁰ = 8^b

 

 

Since the bases are the same in both equations, we just equate the exponents.

 

5a = 15                               10 = b

 

 

We already know b.   Solve for a.

 

 

Answer 2

by the rules of exponents

 

(3^a*8²)⁵=3¹⁵*8^b

 

3^5a*8¹⁰=3¹⁵*8^b

 

you should be able to see the similarities here now.

 

concepts being used

a^xa^y = a^x+y

(a^x)^y = a^xy

a^xb^x = (ab)^x