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.
First, let's apply the properties of integer exponents to rewrite the expression with only positive exponents:
(14^3 * 27^-7)^0 / 3^4
Using the property (a^m)^n = a^(m * n), we can rewrite the expression as:
14^(3 * 0) * 27^(-7 * 0) / 3^4
Simplifying further, we get:
14^0 * 27^0 / 3^4
Any number raised to the power of 0 is equal to 1, so both 14^0 and 27^0 equal 1:
1 * 1 / 3^4
Simplifying further:
1 / 81
Therefore, the simplified fraction with no remaining exponents 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.
Let's apply the properties of integer exponents to generate equivalent expressions with positive exponents:
(4^3 * 7^-2)^-5 / (7^9 * 4^-13)
Using the property (a^m)^n = a^(m * n), we have:
4^(3 * -5) * 7^(-2 * -5) / (7^9 * 4^-13)
Simplifying further:
4^(-15) * 7^10 / (7^9 * 4^-13)
Now, let's apply the property a^m / a^n = a^(m - n) to simplify the expression:
4^(-15 - (-13)) * 7^(10 - 9)
Simplifying the exponent, we have:
4^(-15 + 13) * 7^(1)
Next, applying the property a^(-m) = 1 / a^m, we can rewrite the expression as:
(1 / 4^(15 - 13)) * 7^1
Simplifying the exponent,
(1 / 4^2) * 7^1
Simplifying further,
(1 / 16) * 7
7 / 16
Therefore, the simplified fraction with no remaining exponents is 7/16.
To apply the properties of integer exponents, we can start by simplifying each part of the expression individually:
1. Simplify (14^3):
14^3 = 14 * 14 * 14 = 2744
2. Simplify (27^-7):
To get rid of the negative exponent, we can rewrite it as the reciprocal:
27^-7 = 1 / 27^7 = 1 / (27 * 27 * 27 * 27 * 27 * 27 * 27) = 1 / 387420489
3. Simplify (3^4):
3^4 = 3 * 3 * 3 * 3 = 81
Now, substitute these simplified values back into the original expression:
(14^3 * 27^-7)^(0/3^4) = (2744 * 1 / 387420489)^(0/81)
The expression raised to the power of zero is equal to 1:
(2744 * 1 / 387420489)^(0/81) = 1
Therefore, the simplified value of the expression is 1.
To apply the properties of integer exponents to generate equivalent expressions, we will use the following rules:
1. Product Rule: (a^m * b^n) = a^(m+n)
2. Quotient Rule: (a^m / b^n) = a^(m-n)
3. Power Rule: (a^m)^n = a^(m*n)
Let's break down the given expression step by step:
(14^3 * 27^(-7))^(0/3^4)
Step 1: Simplify the exponents inside the parentheses using the product rule:
14^(3 * (0/3^4)) * 27^(-7 * (0/3^4))
Step 2: Simplify the exponents inside each term:
14^(0/3^3) * 27^(0/3^4)
Step 3: Simplify the exponents further:
14^(0/27) * 27^(0/81)
Step 4: Any number raised to the power of zero is equal to 1, so we substitute:
1 * 1
Step 5: The product of 1 and 1 is still 1, so the final equivalent expression is:
1
Therefore, the simplified fraction is 1/1, which is equal to 1.