Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0/3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(14^3⋅27^−7)^0/3^4 =
To simplify the expression, we can use the property of exponents that states that anything raised to the power of 0 is equal to 1.
Therefore, (14^3⋅27^−7)^0 = 1
Now, we can simplify the remaining part of the expression, which is 1/3^4.
3^4 can be rewritten as (3^2)^2 = 9^2 = 81.
Therefore, (14^3⋅27^−7)^0/3^4 = 1/81.
So, the simplified fraction is 1/81.
Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 =
To simplify the expression, we can use the properties of integer exponents.
First, let's simplify the base (4^3 ⋅ 7^−2):
4^3 = (2^2)^3 = 2^6
7^-2 = 1/7^2 = 1/49
So, (4^3 ⋅ 7^−2)^−5 = (2^6 ⋅ 1/49)^−5 = (2^6/49)^−5 = (2^6)^−5 / (49)^−5 = 2^-30 / 49^-5 = 1/2^30 ⋅ 49^5
Next, let's simplify the expression 7^9 ⋅ 4^-13:
4^-13 = 1/4^13 = 1/2^26
7^9 = (7^3)^3 = 343^3 = 42,949,343
So, 7^9 ⋅ 4^-13 = 42,949,343 ⋅ 1/2^26
Now, let's simplify the entire expression:
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^-13 = (1/2^30 ⋅ 49^5) / (42,949,343 ⋅ 1/2^26)
To simplify the division, we can multiply by the reciprocal:
= (1/2^30 ⋅ 49^5) * (2^26 / 42,949,343)
= 1/(2^30 ⋅ 42,949,343) ⋅ 49^5 ⋅ 2^26
Now, we can write the expression with positive exponents and simplify further:
= 49^5⋅2^26 / (2^30 ⋅ 42,949,343)
Since 49^5 can be expressed as (7^2)^5 = 7^10, we have:
= 7^10 ⋅ 2^26 / (2^30 ⋅ 42,949,343)
Using the property of exponents that states 2^a / 2^b = 2^(a-b):
= 7^10 ⋅ 2^(26-30) / 42,949,343
= 7^10 ⋅ 2^-4 / 42,949,343
Since 2^-4 = 1/2^4 = 1/16, we have:
= 7^10 / (16 ⋅ 42,949,343)
Finally, simplifying further, we have:
= 7^10 / 687,190,948
Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
4^0 ⋅ 5^−2/(−2)^−3=
To simplify the expression, we can use the properties of integer exponents.
First, let's simplify the exponent of 4^0.
Any non-zero number raised to the power of 0 is equal to 1.
So, 4^0 is equal to 1.
Next, let's simplify the exponent of 5^−2.
A negative exponent indicates that we take the reciprocal of the base raised to the positive exponent.
So, 5^−2 is equal to 1/5^2 = 1/25.
Now, let's simplify the exponent of (−2)^−3.
Again, a negative exponent indicates that we take the reciprocal of the base raised to the positive exponent.
So, (−2)^−3 is equal to 1/(-2)^3 = 1/(-8) = -1/8.
Putting it all together, we have:
4^0 ⋅ 5^−2/(−2)^−3 = 1 ⋅ 1/25 / (-1/8)
When we divide by a fraction, we can multiply by the reciprocal of the divisor.
So, the expression becomes:
1 ⋅ 1/25 * (-8/-1)
Multiplying the numerators and denominators, we get:
1/25 * 8/1 = 8/25
Therefore, the simplified value of the expression is 8/25.
To apply the properties of integer exponents and generate equivalent expressions with only positive exponents, we can use the following rules:
1. Any number raised to the power of 0 is equal to 1.
2. The product of two numbers raised to an exponent is equal to each number raised to that exponent, multiplied together.
3. The quotient of two numbers raised to an exponent is equal to each number raised to that exponent, divided by each other.
Applying these rules, we can simplify the expression step by step:
(14^3⋅27^(-7))^0 / 3^4
Step 1: Simplify the exponents within the parentheses.
(14^3⋅(1/27^7))^0 / 3^4
Step 2: Apply the exponent rule for negative exponents to rewrite 27^(-7) as (1/27^7).
(14^3⋅(1/(1/27^7)))^0 / 3^4
Step 3: Simplify the expression within the parentheses.
(14^3⋅27^7)^0 / 3^4
Step 4: Apply the exponent rule for raising a power to the exponent of 0. Any number raised to the power of 0 equals 1.
1 / 3^4
Step 5: Simplify the exponent.
1 / 81
Therefore, the final simplified fraction is 1/81.
To generate an equivalent expression with positive exponents, we can use the properties of integer exponents. Let's break down the steps:
Step 1: Simplify the base and exponent of each term.
- (14^3⋅27^−7) is already simplified because the bases (14 and 27) and exponents (3 and -7) cannot be simplified further.
Step 2: Apply the property of raising a product of terms to an exponent:
- (14^3⋅27^−7)^0 = 14^(3*0) ⋅ 27^(-7*0)
Step 3: Simplify the exponents:
- 14^(3*0) = 14^0 = 1 (Any number raised to the power of 0 is equal to 1)
- 27^(-7*0) = 27^0 = 1 (Any number raised to the power of 0 is equal to 1)
Step 4: Substitute the simplified exponents back into the expression:
- (14^3⋅27^−7)^0 = 1 ⋅ 1 = 1
Now, to solve the expression 1/3^4, we can simplify it further:
Step 5: Simplify the denominator: 3^4 = 3^(2*2) = (3^2)^2 = 9^2 = 81
Step 6: Substitute the simplified denominator back into the expression: 1/81
Therefore, (14^3⋅27^−7)^0/3^4 simplifies to 1/81.