What is the simplified form of;
3a^4b^-2c^3
4, -2, and 3 are powers
A:
81a^4c^3
________
b^2
B:
81a^4
________
b^2c^3
C:
3a^4
________
b^2c^3
D:
3a^4c^3
________
b^2
I looked through my lesson but couldn't find out how to do this... I think its D?
You will need the rule for negative exponents
a^(-x) = 1/a^x
so....
3a^4b^-2c^3
= 3(a^4)(1/b^2)(c^3)
= 3 a^4 c^3 / b^2 , which matches D
To simplify the expression 3a^4b^-2c^3, we can use the rules of exponents.
First, let's simplify the powers of the variables:
- For the variable a, we keep the base "a" and add the exponents: a^4
- For the variable b, since the exponent is -2, we can rewrite it as 1/b^2 (by flipping the base and changing the sign of the exponent).
- For the variable c, we keep the base "c" and add the exponent: c^3
Now, let's substitute the simplified powers back into the expression:
3a^4b^-2c^3 = 3(a^4)(1/b^2)(c^3)
Next, we can multiply the coefficients (numbers) together:
3 * 1 = 3
Now, let's combine the variables with the same base:
a^4 * 1/b^2 * c^3 = a^4c^3/b^2
Therefore, the simplified form of 3a^4b^-2c^3 is:
3a^4c^3/b^2
So the correct answer is D.
To simplify the expression 3a^4b^-2c^3, we need to apply the rules of exponents.
Let's break down the expression step by step:
1. The base numbers are 3, a, b, and c.
2. The exponents are 4, -2, and 3.
Using the rules of exponents:
- When you multiply variables with the same base, you add their exponents.
- When you divide variables with the same base, you subtract their exponents.
Now let's determine the simplified form of the expression:
1. Start with the numerator (the numbers and variables to be multiplied).
a) Simplify a^4 by keeping the base (a) and adding the exponents (4).
The result is a^4.
b) Simplify b^-2 by keeping the base (b) and changing the sign of the exponent (-2) to positive.
When the exponent is negative, it means the reciprocal of the base to the positive exponent.
So, b^-2 = 1 / b^2.
Therefore, the numerator becomes a^4 * 1 / b^2 = a^4 / b^2.
2. Move to the denominator (the numbers and variables to be multiplied) and simplify.
Simplify c^3 by keeping the base (c) and the exponent (3).
Therefore, the denominator becomes c^3.
3. The final simplified form is (a^4 / b^2c^3).
Remember to write the terms in the correct order.
Now, let's compare the simplified form with the answer choices:
A: 81a^4c^3 / b^2
This answer choice does not match the simplified form because of the 81.
So, A is not the correct choice.
B: 81a^4 / b^2c^3
This answer choice matches the simplified form.
So, B is the correct choice.
C: 3a^4 / b^2c^3
This answer choice is missing the factor of 3 in the numerator.
So, C is not the correct choice.
D: 3a^4c^3 / b^2
This answer choice has the numerator and denominator switched.
So, D is not the correct choice.
Therefore, the correct choice is B: 81a^4 / b^2c^3.