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.
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
3 answers
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.
(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.
3 answers