Rules of exponents

Name: Carissa

Who is asking: Student
Level: Middle

Question: how do you work this out? Investigate the relationship between a,b,c and d if 2^a*2^b=4^c/4^d?

Hi Carissa, Hi Carissa,

To find the relationship between a, b, c and d, you must first know the rules of exponents:

When you multiply two numbers with the same base, you add the exponents

eg. a² * a³ = a²⁺³ = a⁵
When you divide two numbers with the same base, you subtract the exponentÊof the number in the denominator from the exponent of the number in the numerator.

eg. a⁵/a² = a^5-2 = a³
When you raise a power to another power, you multiply the exponents

eg. (a³)² = a^2*3 = a⁶

You will need to apply all three of these rules to find the relationship between a, b, c and d in your problem:

2^a * 2^b = 4^c/4^d		This is given. First apply rule 1 to left-hand side
2^a+b = 4^c/4^d		Now apply rule 2 to right-hand side
2^a+b = 4^c-d		We need to have the same base on both sides, so re-write 4 as 2²
2^a+b = (2²)^c-d		Now apply rule 3 to the right-hand side
2^a+b = 2^2(c-d)		Since we have the same base on both sides (i.e. 2), and the right-hand side equals the left-hand side, we know that the exponents must be equal

Therefore, a + b = 2(c - d) This is the relationship between a, b,Êc and d

Cheers,

Leeanne