Question

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.

Answer 1

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 $\frac{2744}{81}$.