Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27^−7)^0 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.?
To apply the properties of integer exponents, we can use the rule that $(a^m \cdot b^n)^p = a^{mp} \cdot b^{np}$.
Using this rule, we can rewrite $(14^3⋅27^{-7})^0$ as $14^{3 \cdot 0} \cdot 27^{-7 \cdot 0}$.
This simplifies to $14^0 \cdot 27^0$, since any number raised to the power of 0 is equal to 1.
Therefore, $(14^3⋅27^{-7})^0 = 1$.
thats wrong
I apologize for the confusion. You are correct.
To generate equivalent expressions with only positive exponents, we need to use the properties of integer exponents.
First, let's simplify the exponents:
$14^3 = 14 \cdot 14 \cdot 14$
$27^{-7} = \frac{1}{27^7}$ (using the property $a^{-n} = \frac{1}{a^n}$)
Now, let's substitute these values back into the expression:
$(14^3 \cdot 27^{-7})^0 = (14 \cdot 14 \cdot 14 \cdot \frac{1}{27^7})^0$
Applying the power of zero property, any non-zero number raised to the power of zero is equal to 1, the expression simplifies to:
$1^0 = 1$
Therefore, $(14^3 \cdot 27^{-7})^0 = 1$.
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.?
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents:
First, let's simplify the exponents and rewrite the expression:
$14^3 = 14 \cdot 14 \cdot 14$
$27^{-7} = \frac{1}{27^7}$
$3^4 = 3 \cdot 3 \cdot 3 \cdot 3$
Substituting these values into the expression:
$(14^3 \cdot 27^{-7})^0/3^4 = (14 \cdot 14 \cdot 14 \cdot \frac{1}{27^7})^0/(3 \cdot 3 \cdot 3 \cdot 3)$
Now, let's apply the rule that $(a^m \cdot b^n)^p = a^{mp} \cdot b^{np}$:
$(14 \cdot 14 \cdot 14 \cdot \frac{1}{27^7})^0/(3 \cdot 3 \cdot 3 \cdot 3) = (14^{1 \cdot 3} \cdot \frac{1}{27^{7 \cdot 1}})^0/(3^{1 \cdot 4})$
Using the power of zero property, the numerator becomes 1:
$1/(3^{4})$
Simplifying the expression:
$1/81$
Therefore, $(14^3 \cdot 27^{-7})^0/3^4 = 1/81$.
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.
To generate equivalent expressions with only positive exponents, we can use the properties of integer exponents:
First, let's simplify the exponents and rewrite the expression:
$4^3 = 4 \cdot 4 \cdot 4$
$7^{-2} = \frac{1}{7^2}$
$7^9 = 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7$
$4^{-13} = \frac{1}{4^{13}}$
Next, let's substitute these values back into the expression:
$(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot 4^{-13} = (4 \cdot 4 \cdot 4 \cdot \frac{1}{7^2})^{-5}/(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7) \cdot \frac{1}{4^{13}}$
Now, let's apply the properties of exponents:
$(4 \cdot 4 \cdot 4 \cdot \frac{1}{7^2})^{-5}/(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7) \cdot \frac{1}{4^{13}} = \frac{(4^3)^{-5} \cdot (7^{-2})^{-5}}{(7^9 \cdot 4^{-13})}$
Using the rules for exponents, let's simplify the numerator:
$(4^3)^{-5} = 4^{-15}$
$(7^{-2})^{-5} = 7^{10}$
Now, let's substitute these values back into the expression:
\frac{4^{-15} \cdot 7^{10}}{7^9 \cdot 4^{-13}} = \frac{4^{-15} \cdot 7^{10}}{\frac{1}{7^{-9}} \cdot \frac{1}{4^{13}}} =4^{-15} \cdot 7^{10} \cdot 7^9 \cdot 4^{13}
Using the rule that $a^m \cdot a^n = a^{m+n}$, we simplify as follows:
4^{-15} \cdot 7^{10} \cdot 7^9 \cdot 4^{13} = 4^{-2} \cdot 7^{19}
Finally, we can rewrite this expression as a simplified fraction:
\frac{1}{4^2} \cdot 7^{19} = \frac{1}{16} \cdot 7^{19} = \frac{7^{19}}{16}
Therefore, the simplified expression is $\frac{7^{19}}{16}$.