what is an equivalent expression to (2^-3*9^3)-4/2^9*9^-10 with only positive exponents generated by applying the properties of integer exponents
To simplify the expression and rewrite it with only positive exponents, we can apply the properties of integer exponents, specifically:
1. The property a^(-n) = 1/a^n
2. The property (a/b)^n = a^n / b^n
3. The property a^m * a^n = a^(m + n)
4. The property a^m / a^n = a^(m - n)
Now let's simplify step by step:
(2^-3 * 9^3) - (4/2^9 * 9^-10)
= (1/2^3 * 9^3) - (4/2^9 * 1/9^10) [Applying property 1]
= (1/(2^3))^3 * (9^3) - (4/(2^9)) * (1/(1/(9^10))) [Applying property 2]
= (1/8)^3 * (9^3) - (4/(2^9)) * (9^10) [Simplifying]
= 1/512 * 729 - (4/512) * 3486784401 [Applying property 3]
= 729/512 - 13947137604/512
= (729 - 13947137604) / 512
= -13947136875 / 512
Therefore, the equivalent expression with only positive exponents is: -13947136875/512.
simplify that
Apologies for the mistake in the previous response. Let's simplify the expression correctly:
(2^-3 * 9^3) - (4/2^9 * 9^-10)
Using the properties of integer exponents:
= (1/(2^3) * 9^3) - (4/(2^9) * 1/(9^10)) [Applying property 1]
= (1/(8) * 9^3) - (4/(512) * 1/(9^10))
= (9^3/8) - (4/(512 * 9^10)) [Consolidating fractions]
= (9^3/8) - (4/((2^9) * (9^10))) [Rewriting 512 as 2^9]
= (729/8) - (4/(512 * (9^10))) [Expanding the exponents]
= (729/8) - (4/(512 * (9^10)))
Now, let's simplify further:
= (729/8) - (4/(512 * (3^20))) [Expressing 9^10 as (3^2)^10 = 3^20]
= (729/8) - (4/(512 * 3^20))
= (729/8) - (4/(512 * 3486784401)) [Evaluating 3^20 as 3486784401]
= (729/8) - (4/1789569703424)
To find a common denominator, we can multiply the first fraction by (221196550428) / (221196550428):
= (729/8) * (221196550428/221196550428) - (4/1789569703424)
= (729 * 221196550428) / (8 * 221196550428) - (4/1789569703424)
= 159922120432012 / 1769572403424 - (4/1789569703424)
= 159922120432012 / 1769572403424 - 4/1789569703424
Now, we can combine the fractions:
= (159922120432012 - 4) / 1769572403424
= 159922120432008 / 1769572403424
The simplified positive exponent expression is: 159922120432008 / 1769572403424.