Apply the Properties of integer exponents to generate an equivalent expression to (14^3x27^(-7)^0/3^4 with only positive exponents, then solve the expression. As a simplified fraction.
To apply the properties of integer exponents, we can use the following rules:
1. $a^m \cdot a^n = a^{m+n}$
2. $\frac{a^m}{a^n} = a^{m-n}$
3. $(a^m)^n = a^{m \cdot n}$
Now, let's simplify the expression step by step:
First, let's apply the exponent rule to simplify $14^3$:
$14^3 = 14 \cdot 14 \cdot 14 = 2744$
Next, let's apply the exponent rule to simplify $27^{-7}$:
$27^{-7} = \frac{1}{27^7} = \frac{1}{(3^3)^7} = \frac{1}{3^{3 \cdot 7}} = \frac{1}{3^{21}}$
Now, let's apply the exponent rule to simplify $3^4$:
$3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$
Putting it all together, the expression becomes:
$\frac{2744 \cdot 1}{81} = \frac{2744}{81}$
To simplify this fraction, we can find the greatest common divisor (GCD) and divide both the numerator and denominator by it. The GCD of 2744 and 81 is 1, so the fraction is already in its simplest form.
Therefore, the simplified fraction of the expression is $\frac{2744}{81}$.
To apply the properties of integer exponents and generate an equivalent expression with only positive exponents, we can use the following rules:
1. Product Rule: When multiplying two numbers with the same base, add their exponents.
2. Quotient Rule: When dividing two numbers with the same base, subtract the exponent of the denominator from the exponent of the numerator.
3. Power Rule: When raising a power to another power, multiply the exponents.
Let's break down the given expression step-by-step:
Step 1: Expand the exponents using the rules above:
(14^3 * 27^(-7))^0 / 3^4
Step 2: Applying the product and power rules:
(14^3)^0 * (27^(-7))^0 / 3^4
Step 3: Any number raised to the power of 0 is always 1:
1 * 1 / 3^4
Step 4: Evaluate 3^4:
1 / 81
Therefore, the equivalent expression with only positive exponents is 1/81.
To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents. Let's break down the given expression and simplify it step by step:
(14^3 * 27^(-7)^0) / 3^4
Step 1: Simplify the exponent 27^(-7)^0
Any number raised to the power of 0 is always equal to 1.
Therefore, 27^(-7)^0 = 27^(-7 * 0) = 27^0 = 1.
Now our expression becomes:
(14^3 * 1) / 3^4
Step 2: Simplify the expression inside the parentheses.
14^3 = 14 * 14 * 14 = 2,744.
So the expression becomes:
(2,744 * 1) / 3^4
Step 3: Simplify the exponent 3^4.
3^4 = 3 * 3 * 3 * 3 = 81.
Now the expression becomes:
(2,744 * 1) / 81
Step 4: Solve the expression.
To divide by 81, we can rewrite it as a fraction.
2,744 / 81 can be written as the fraction 2744/81.
Therefore, the simplified fraction equivalent to the initial expression is 2744/81.
Let me know if I can help you with anything else.