Apply the Properties of Integer Exponents to generate an equivalent expression to (5−3)6⋅ 51710−2
with only positive exponents, then solve the expression. (1 point)
(5^−3)^6 x 5^17 /10^−2=
To apply the properties of integer exponents, we multiply the exponents when raising an exponential expression to a power.
So we have:
(5^-3)^6 x 5^17 /10^-2
The exponent outside the parentheses is multiplied by the exponents inside the parentheses:
5^(-3*6) x 5^17 /10^-2
= 5^-18 x 5^17 /10^-2
Since the bases are the same (5) and we are dividing, we subtract the exponents:
= 5^(17 - 18) / 10^-2
= 5^(-1) / 10^-2
To make the exponent positive, we can rewrite it as the reciprocal:
= 1/5^1 / 1/10^2
= 10^2 / 5^1
= 100 / 5
= 20
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents:
1. For any nonzero number a, a^0 = 1.
2. For any nonzero number a, a^-n = 1/a^n.
Using these properties, we can rewrite the expression as follows:
(5^(-3))^6 * 5^17 / 10^(-2)
= (1/5^3)^6 * 5^17 / (1/10^2)
= (1/125)^6 * 5^17 / (1/100)
= (1/244140625) * 5^17 / (1/100)
Next, we can simplify the expression by multiplying the terms:
= 5^17 * (1/244140625) * 100
= (5^17 * 100) / 244140625
Now, let's solve the expression:
= (762939453125 * 100) / 244140625
= 76293945312500 / 244140625
= 312500
Therefore, the solution is 312500.
To apply the properties of integer exponents, we can rewrite the expression (5^(-3))^6 as (5^(-3*6)). According to the rule of multiplying exponents with the same base, this can be further simplified as 5^(-18). Similarly, we can rewrite 5^17/10^(-2) as (5^17)*(10^2). Now, let's solve the expression step by step:
(5^(-3))^6 x 5^17 /10^(-2)
= (5^(-18)) x (5^17) x (10^2)
To multiply powers with the same base, we add the exponents:
= 5^(-18 + 17) x 10^2
= 5^(-1) x 10^2
Since 5^(-1) is the same as 1/5, we have:
= (1/5) x 10^2
= 1/5 x 100
To multiply fractions, we multiply the numerators and denominators:
= 100/5
= 20
Therefore, the equivalent expression with only positive exponents is 20, and the solved expression is 20.