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)
To apply the properties of integer exponents, we need to remember the rules:
1. Product of powers: (a^m) * (a^n) = a^(m+n)
2. Quotient of powers: (a^m) / (a^n) = a^(m-n)
3. Power of a power: (a^m)^n = a^(m*n)
4. Power of a product: (ab)^n = a^n * b^n
5. Power of a quotient: (a/b)^n = a^n / b^n
Using these rules, we can simplify the expression:
(14^3 * 27^(-7))^0 / 3^4
First, let's simplify the exponents of 14 and 27:
14^3 = 14 * 14 * 14 = 2744
27^(-7) = 1/27^7 = 1/(3^(-3))^7 = 1/3^(-21) = 3^21
Now, we can substitute these values back into the expression:
(2744 * 3^21)^0 / 3^4
Anything raised to the power of 0 is equal to 1, so we have:
1 / 3^4
And 3^4 is equal to 81, so the final simplified expression is:
1 / 81
Therefore, the solution is 1/81.
Well, isn't this an exponent extravaganza! Let's break it down, shall we?
First, we have (14^3⋅27^−7)^0/3^4. According to the properties of integer exponents, any number raised to the power of 0 is equal to 1. So, we can simplify this expression by replacing (14^3⋅27^−7)^0 with 1.
Now, we're left with 1/3^4. To make all our exponents positive, we can rewrite 1 as 1^1. So, the expression becomes (1^1)/(3^4).
Using the property (a^m)/(a^n) = a^(m-n), we can simplify this further to 1^(1-4) / 3^4, which gives us 1^-3 / 3^4.
Since any number raised to the power of -1 is its reciprocal, we can write this as 1 / (1 * 3^4).
Finally, evaluating this expression, we have 1 / (1 * 81), which simplifies to 1/81.
So, the solution to the expression (14^3⋅27^−7)^0/3^4 is a delightful fraction: 1/81.
To apply the properties of integer exponents, we can 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 apply these rules step by step to simplify the expression:
Step 1: Simplify the exponent inside the parentheses:
(14^3 ⋅ 27^(-7))^0 / 3^4
= (14^3 ⋅ (1/27^7))^0 / 3^4
= (14^3 / 27^7)^0 / 3^4
Step 2: Apply the power rule to the numerator:
(14^3 / 27^7)^0 / 3^4
= (14^(3*0) / 27^(7*0)) / 3^4
= (14^0 / 27^0) / 3^4
Step 3: Simplify the exponents of 14 and 27:
(14^0 / 27^0) / 3^4
= (1 / 1) / 3^4
= 1 / 3^4
= 1 / (3^2 * 3^2)
= 1 / 9
Therefore, the expression (14^3⋅27^−7)^0/3^4 simplifies to the fraction 1/9.
To apply the properties of integer exponents, we can use the following rules:
1. Product Rule: (a^m)(a^n) = a^(m+n)
2. Quotient Rule: (a^m)/(a^n) = a^(m-n)
3. Power Rule: (a^m)^n = a^(m*n)
First, let's simplify the expression inside the parentheses: (14^3⋅27^−7)
To simplify the expression, we can apply the product rule and the power rule:
14^3 = 14 * 14 * 14 = 2,744
27^(-7) = 1/(27^7) = 1/((3^3)^7) = 1/3^(3*7) = 1/3^21
Now, substituting these values back into the original expression:
(14^3⋅27^−7)^0/3^4 = (2,744 * (1/3^21))^0/3^4
Next, we can simplify the expression further by applying the power rule again:
(2,744 * (1/3^21))^0/3^4 = (2,744^0 * (1/3^21)^0)/3^4
According to the power rule, any number raised to the power of 0 is equal to 1. Therefore:
(2,744^0 * (1/3^21)^0)/3^4 = (1 * 1)/3^4 = 1/3^4
Finally, let's simplify the fraction by evaluating 3^4:
1/3^4 = 1/81
So the solution to the expression is 1/81.