Simplification

Question

What is the best possible way to simplify these?

 

(4a^-2b)² (3a³b^-4)^-2

 

(125/27)^-2/3

 

 

Answer 1

For the second one, we can use the rules of exponents.  Apply the rule

 

(x / y)ⁿ = xⁿ / yⁿ

 

 

First, rewrite 27 and 125 using bases raised to exponent.

 

27 = 3³

125 = 5³

 

 

Then,

 

(27 / 125)^-2/3 =

 

(3³)^-2/3 / (5³)^-2/3

 

 

Multiply the exponents and keep the base for each term.

 

3^-2 / 5^-2 =

 

(1/9) / (1/25) =

 

25/9

Answer 2

For the first problem, remember the rules of exponents. When you multiply like bases, you add the exponents. When you have an expression in parentheses raised to a power, you multiply the exponents.

 

For instance, (4a^-2b)² = 4² * (a^-2)² * b² = 16 * a^{-2 * 2} * b² = 16a^-4b²

 

Can you do the second term, Ken?

 

As for the second problem, whenever we have negative exponents, we find the reciprocal of the fraction and raise the denominator to the positive power.

 

In other words, (125/27)^-2/3 = (27/125)^2/3

 

Now recall what fractional exponents mean. The denominator means to take the root of the base, and the numerator means to raise the base to that power.

 

In our problem, we need to take the cube root of 27/125, and then raise that result to the second power.

 

Can you handle that?