Simplify the following expression, and write your answer using only positive exponents.
(-2r^-16)^-3
(The answer is not -8r^48)
Since:
a ⁻ ⁿ = 1 / aⁿ
(- 2 r ⁻ ¹⁶ ) ⁻ ³ = 1 / ( - 2 r ⁻ ¹⁶ )³ = 1 / [ ( - 2 )³ ∙ ( r ⁻ ¹⁶ )³ ] = 1 / ( - 8 r ⁻ ⁴⁸ ) =
( 1 / - 8 ) ∙ 1 / r ⁻ ⁴⁸ = - 1 / 8 r ⁴⁸
To simplify the expression (-2r^-16)^-3 and write the answer using only positive exponents, we can follow the order of operations and apply the exponent rule for negative exponents.
First, let's simplify the negative exponent within the parentheses. The exponent "-16" means we have to take the reciprocal of the base "r". So, -2r^-16 becomes -2/r^16. Now, we have:
(-2/r^16)^-3
To simplify this further, we can apply the rule that says when we raise a fraction to a negative exponent, we can flip the fraction and change the exponent to positive.
Therefore, (-2/r^16)^-3 can be rewritten as (r^16/-2)^3.
Expanding this expression, we get:
(r^3)^16/(-2)^3
Now, we simplify the positive exponent by raising the base and the exponent:
r^48/(-2)^3
The cube of -2 is -8, so we have:
r^48/-8
Finally, to write the answer using only positive exponents, we can rewrite -8 as (-1) * 2^3:
r^48/(-1 * 2^3)
This can be simplified further as:
r^48/(-8)
Since the denominator is negative, we can bring the negative sign to the numerator, resulting in:
-r^48/8
Therefore, the simplified expression with positive exponents is -r^48/8.
To simplify the expression (-2r^-16)^-3 using only positive exponents, you can follow these steps:
Step 1: Apply the negative exponent to the base inside the parentheses.
(-2^-1 * r^16)^-3
Step 2: Simplify the negative exponent.
((-1/2) * r^16)^-3
Step 3: Apply the negative exponent to each term inside the parentheses.
(-1/2)^-3 * (r^16)^-3
Step 4: Simplify each term raised to a negative exponent.
(-2/1)^3 * r^(16 * -3)
Step 5: Simplify the exponents.
(-8) * r^-48
Thus, the simplified expression with positive exponents is -8r^-48.