which expression is equivalent to (20a^2b^0c^-4) (3a^15b^12c^0)
for all values of a, b, and c where the expression is defined?
A 60a^30
B 23a^17b^22/c^4
C 60a^17b^12/c^4
D23a^15b^12c^-4
The answer is option C: 60a^17b^12/c^4.
To simplify this expression, we need to add the exponents of like terms (same variables). For a, we get 20a^2 * 3a^15 = 60a^17. For b, we get b^0 (which is 1) times b^12 = b^12. For c, we get c^-4 * c^0 (which is 1) = c^-4.
Putting it all together, we get 60a^17b^12/c^4, which is equivalent to the given expression.
To simplify the given expression (20a^2b^0c^-4) (3a^15b^12c^0), you need to apply the rules of exponents.
First, let's simplify the numbers: 20 multiplied by 3 equals 60. We can remove the coefficient and focus on the variables.
Next, let's simplify the variables with exponents. Recall the rules of exponents:
1. When multiplying variables with the same base, you add the exponents.
2. When dividing variables with the same base, you subtract the exponents.
3. Any variable raised to the power of 0 is equal to 1.
Now let's simplify the variables step by step:
a^2 * a^15 = a^(2+15) = a^17
b^0 * b^12 = b^(0+12) = b^12
c^-4 * c^0 = c^(-4+0) = c^-4 (using rule 3)
Combining all the simplified variables, we have:
60a^17b^12c^-4
Now we need to check the given options to find an equivalent expression. Let's evaluate each option:
A 60a^30: This is not equivalent to the original expression since it has a different exponent for the variable 'a.'
B 23a^17b^22/c^4: This is not equivalent to the original expression since it differs in the variables 'b' and 'c.'
C 60a^17b^12/c^4: This is equivalent to the original expression because it has the same exponents for all three variables, 'a,' 'b,' and 'c.' It simplifies to the same expression obtained above.
D 23a^15b^12c^-4: This is not equivalent to the original expression since it has different exponents for the variables 'a' and 'c.'
Therefore, the correct answer is Option C: 60a^17b^12/c^4.
To simplify the expression (20a^2b^0c^-4) (3a^15b^12c^0), we can simplify the numerical coefficients and combine the variables with the same bases.
Step 1: Simplify the numerical coefficients.
20 * 3 = 60
Step 2: Combine variables with the same bases.
a^2 * a^15 = a^(2 + 15) = a^17
b^0 * b^12 = b^(0 + 12) = b^12
c^-4 * c^0 = c^(-4 + 0) = c^-4
Putting it all together, the simplified expression is:
60a^17b^12c^-4
The equivalent expression is option C) 60a^17b^12/c^4.