Apply the properties of integer exponents to simplify the algebraic expression (-5a^3b^4c^-2/ 3a^-4b^0c^3)^-2
To simplify the given algebraic expression, (-5a^3b^4c^-2/3a^-4b^0c^3)^-2, we can apply the properties of integer exponents:
1) A negative exponent indicates the reciprocal of the base raised to the positive exponent.
2) Any base raised to the power of 0 equals 1.
Let's simplify the expression step by step:
(-5a^3b^4c^-2/3a^-4b^0c^3)^-2
First, deal with the negative exponents:
(-5a^3b^4c^-2/3 * 1/(a^-4) * 1/(b^0) * (c^3))^-2
Next, simplify the expressions with negative exponents by taking their reciprocal:
(-5a^3b^4/c^2 * a^4/3 * 1/(c^3))^-2
Now simplify the remaining exponents by multiplying:
(-5a^7b^4/3c^5)^-2
To get rid of the negative sign, we can take the reciprocal of the entire expression:
[1/(-5a^7b^4/3c^5)]^2
Simplify the expression inside the reciprocal:
[3c^5/(-5a^7b^4)]^2
Now, square each term within the brackets:
(3^2 * c^(5*2))/((-5)^2 * a^(7*2) * b^(4*2))
Simplify inside the brackets:
(9c^10)/(25a^14b^8)
Therefore, the simplified form of the given expression is (9c^10)/(25a^14b^8).
To simplify the algebraic expression (-5a^3b^4c^-2/ 3a^-4b^0c^3)^-2, we'll apply the properties of integer exponents and work step by step:
Step 1: Deal with the negative exponent:
Recall that any expression with a negative exponent can be rewritten by moving the base to the denominator and changing the sign of the exponent.
So, we can rewrite c^-2 as 1/c^2.
The expression now becomes: (-5a^3b^4/3a^-4b^0c^3)^-2 * 1/c^2
Step 2: Simplify b^0:
Any number or variable raised to the power of zero equals 1. Therefore, b^0 can be simplified to 1.
The expression now becomes: (-5a^3/3a^-4c^3)^-2 * 1/c^2
Step 3: Simplify the negative exponent:
When we have an expression raised to a negative exponent, we can rewrite it by moving the entire expression to the denominator and changing the sign of the exponent.
So, (-5a^3/3a^-4c^3)^-2 becomes 1/(-5a^3/3a^-4c^3)^2.
The expression now becomes: 1/(-5a^3/3a^-4c^3)^2 * 1/c^2
Step 4: Simplify the exponents inside the parentheses:
To simplify the expression inside the parentheses, we need to apply the exponent properties. When we raise a fraction to a power, we raise both the numerator and denominator to that power.
So, (-5a^3/3a^-4c^3)^2 becomes ((-5)^2 * (a^3)^2) / ((3a^-4)^2 * (c^3)^2).
The expression now becomes: 1/((-5)^2 * (a^3)^2 / (3a^-4)^2 * (c^3)^2) * 1/c^2
Step 5: Simplify the exponents:
Let's simplify the exponents. Remember that when we raise a power to another power, we multiply the exponents.
((-5)^2 * (a^3)^2) becomes (25 * a^6).
((3a^-4)^2 * (c^3)^2) becomes (9 * a^-8 * c^6).
The expression now becomes: 1/(25 * a^6 / 9 * a^-8 * c^6) * 1/c^2
Step 6: Simplify and combine the terms:
To divide by a fraction, we can multiply by its reciprocal:
1/(25 * a^6 / 9 * a^-8 * c^6) is the same as 1 * (9 * a^-8 * c^6) / (25 * a^6).
The expression now becomes: (9 * a^-8 * c^6) / (25 * a^6) * 1/c^2
Step 7: Simplify and combine the terms:
To multiply variables with the same base, we add their exponents.
a^-8 * a^6 becomes a^(-8 + 6), which simplifies to a^-2.
The expression now becomes: (9 * a^-2 * c^6) / (25 * a^6) * 1/c^2
Step 8: Simplify and combine the terms:
To divide variables with the same base, we subtract their exponents.
c^6 / c^2 becomes c^(6-2), which simplifies to c^4.
The expression now becomes: (9 * a^-2 * c^4) / (25 * a^6)
Finally, we simplify further by using negative exponents:
Since a^-2 is in the denominator, we can rewrite it as 1/a^2.
The expression is now: 9 * c^4 / (25 * a^6 * a^2)
And that is the simplified form of the algebraic expression (-5a^3b^4c^-2/ 3a^-4b^0c^3)^-2.