Question

Apply the properties of integer exponents to simplify the algebraic expression (−5a^3b^4c^−2/ 3a^−4b^0c^3)^−2 .

Answer 1

To simplify the given expression, let's apply the properties of integer exponents step by step.

First, let's simplify the expression inside the parentheses:
(−5a^3b^4c^−2/ 3a^−4b^0c^3)

Applying the property of division of exponents, we subtract the exponents with the same base:
= (−5a^3b^4c^(-2+2))/ 3a^(-4)b^0c^3
= (−5a^3b^4c^0)/ 3a^(-4)c^3

Since any number raised to the power of 0 is 1, we can simplify further:
= (−5a^3b^4)/ 3a^(-4)c^3

Now, let's simplify the entire expression by raising it to the power of -2:
(−5a^3b^4c^−2/ 3a^−4b^0c^3)^−2
= 1 / (−5a^3b^4/ 3a^(-4)c^3)^2

Next, let's apply the property of negative exponents:
= 1 / ((3/(-5))a^3a^4b^4c^3)^2

Simplifying the constant:
= 1 / ((-3/5)a^7b^4c^3)^2

Finally, let's apply the property of exponents to square the expression:
= 1 / ((-3/5)^2a^(7*2)b^(4*2)c^(3*2))
= 1 / ((9/25)a^14b^8c^6)

Therefore, the simplified form of the expression is 1 / ((9/25)a^14b^8c^6).