Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(43 ⋅ 7−2)−579 ⋅ 4−13=
We can apply the properties of integer exponents to rewrite the expression with positive exponents.
Start by applying the negative exponent property:
43 ⋅ 7^(-2) = 43 / 7^2
4^(-13) = 1 / 4^13
So, the expression becomes:
(43 / 7^2)^(-579) ⋅ (1 / 4^13)
Next, apply the power of a quotient property:
(43 / 7^2)^(-579) = (43)^(-579) / (7^2)^(-579)
= (43)^(-579) / 7^(-1158)
Now, apply the power of a power property:
7^(-1158) = (7^2)^(-579)
= 49^(-579)
So, the expression is simplified to:
(43)^(-579) / 49^(-579) ⋅ (1 / 4^13)
Now, we can multiply the numerators and denominators separately:
(43)^(-579) ⋅ 1 = (43)^(-579)
49^(-579) ⋅ (4^13) = (49 / 4)^(-579)
Combine the terms:
( (43)^(-579) ) / ( (49 / 4)^(-579) )
To simplify further, we can apply the power property:
( (43)^(-579) ) / ( (49 / 4)^(-579) ) = ( (43) / (49 / 4) )^(-579)
To solve the expression, we need to simplify the fraction:
( (43) / (49 / 4) ) = (43) * (4 / 49)
= (43 * 4) / 49
= 172 / 49
Therefore, the simplified fraction is 172/49.
Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.
Property 1: a^(-n) = 1/a^n
Property 2: (a^m)^n = a^(m*n)
Using these properties, we can rewrite the expression (43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13) as:
((43 ⋅ 1/7^2)^(-579)) ⋅ 1/4^13
Simplifying further, we have:
((43/49)^(-579)) ⋅ 1/4^13
Now, let's solve the expression:
((43/49)^(-579)) ⋅ (1/4^13)
Using the property (a^m)^n = a^(m*n), we can rewrite it as:
(43/49)^(579 * 13)
Since the exponent is positive, we don't need to take the reciprocal. Now, we calculate:
(43/49)^(579 * 13) ≈ (43/49)^7517
Unfortunately, this cannot be simplified further.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents. The following properties will be helpful:
1. Product Rule: a^m * a^n = a^(m+n)
2. Quotient Rule: a^m / a^n = a^(m-n)
3. Power of a Power Rule: (a^m)^n = a^(m*n)
Now let's simplify the given expression step by step:
(43 ⋅ 7^(-2))^(-579) ⋅ 4^(-13)
First, apply the power of a power rule to the innermost expression:
43^(-579) ⋅ (7^(-2))^(-579) ⋅ 4^(-13)
Next, apply the product rule:
43^(-579) ⋅ 7^((-2)*(-579)) ⋅ 4^(-13)
Simplify the exponents:
43^(-579) ⋅ 7^1158 ⋅ 4^(-13)
Now let's focus on the bases. Recall that 43 and 4 have positive exponents, but 7 has a negative exponent. To convert the negative exponent of 7 to a positive exponent, we can apply the quotient rule:
43^(-579) ⋅ (1 / 7^(1158)) ⋅ 4^(-13)
Note that (1/7^1158) can be rewritten as 7^(-1158) using the definition of a negative exponent.
Applying the quotient rule and now all exponents are positive:
43^(-579) ⋅ 7^(-1158) ⋅ 4^(-13)
To simplify the expression further, we can combine the bases by applying the product rule:
(43⋅7) ^ (-579) ⋅ 4^(-13)
Now, we can calculate the values:
43⋅7 = 301
So, the equivalent expression is:
301^(-579) ⋅ 4^(-13)
To solve the expression, we need the values of 301^(-579) and 4^(-13). Calculating these values gives us:
301^(-579) ≈ 2.42 × 10^(-224)
4^(-13) ≈ 1.22 × 10^(-8)
Multiplying these two values, we get:
(2.42 × 10^(-224)) * (1.22 × 10^(-8)) ≈ 2.95 × 10^(-232)
Therefore, the simplified fraction form of the expression is approximately 2.95 × 10^(-232).