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 $\frac{7^{19}}{16}$.
{7^{19}}{16} is 7^19/16?
Yes, you are correct. My apologies for the error. The simplified expression is indeed $\frac{7^{19}}{16}$, which means $7^{19}$ divided by 16. Thank you for catching that mistake.
its wrong
Apologies for the confusion. Let's correct the mistake.
The expression we need to simplify is:
$\frac{7^{19}}{16}$
To simplify this expression and eliminate the exponent, we can rewrite 16 as $2^4$.
Now our expression becomes:
$\frac{7^{19}}{2^4}$
Using the rule for dividing numbers with the same base, we can subtract the exponents:
$7^{19-4}/2^4$
Simplifying further:
$7^{15}/2^4$
Now, to eliminate the exponent for both bases, we can rewrite $2^4$ as 16, and $7^{15}$ remains the same:
$\frac{7^{15}}{16}$
So, the simplified expression is $\frac{7^{15}}{16}$.
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.
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 = ???
To generate equivalent expressions with positive exponents, we will use the properties of integer exponents:
First, let's simplify the exponents and rewrite the expression:
$4^3 = 4 \cdot 4 \cdot 4$
$7^{-2} = \frac{1}{7^2}$
$7^9 = 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7$
$4^{-13} = \frac{1}{4^{13}}$
Now, let's substitute these values back into the expression:
$(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot 4^{-13} = (4 \cdot 4 \cdot 4 \cdot \frac{1}{7^2})^{-5}/(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7) \cdot \frac{1}{4^{13}}$
Next, let's apply the properties of exponents:
$(4 \cdot 4 \cdot 4 \cdot \frac{1}{7^2})^{-5}/(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7) \cdot \frac{1}{4^{13}} = \frac{(4^3)^{-5} \cdot (7^{-2})^{-5}}{(7^9 \cdot 4^{-13})}$
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 $\frac{7^{19}}{16}$.
what does {7^{19}}{16} mean?
I apologize for the confusion. It appears that I made a typo in the previous response. The expression `{7^{19}}{16}` does not have any mathematical meaning.
The correct expression we arrived at is $\frac{7^{19}}{16}$, which represents the division of $7^{19}$ by 16.
but that wrong
Apologies for the previous incorrect response. Let's correct the mistake.
The expression we need to simplify is:
$(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot 4^{-13}$
To simplify this expression and eliminate the negative exponents, we can use the fact that $a^{-n} = \frac{1}{a^n}$.
Rewriting the expression, we have:
$\frac{1}{(4^3 \cdot 7^2)^5} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$
Now, let's simplify the exponents and multiply the terms:
$\frac{1}{(64 \cdot 49)^5} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$
Simplifying further:
$\frac{1}{(3136)^5} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$
Now, let's evaluate $3136^5$:
$3136^5 = (56^2)^5 = 56^{10} = (7 \cdot 8)^{10} = 7^{10} \cdot 8^{10}$
Substituting this back into the expression:
$\frac{1}{7^{10} \cdot 8^{10}} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$
Next, we can simplify $8^{10}$ and $4^{13}$:
$8^{10} = (2^3)^{10} = 2^{30}$
$4^{13} = (2^2)^{13} = 2^{26}$
Substituting these back into the expression:
$\frac{1}{7^{10} \cdot 2^{30}} \cdot \frac{1}{7^9} \cdot \frac{1}{2^{26}}$
To simplify the expression, let's combine the terms with the same base:
$\frac{1}{7^{10+9} \cdot 2^{30+26}}$
Simplifying the exponents:
$\frac{1}{7^{19} \cdot 2^{56}}$
Therefore, the simplified expression is $\frac{1}{7^{19} \cdot 2^{56}}$.
what does $\frac{1}{7^{19} \cdot 2^{56}}$ mean tho???
I apologize for the confusion. Let's clarify the meaning of the expression.
The expression $\frac{1}{7^{19} \cdot 2^{56}}$ represents the reciprocal of the product of $7^{19}$ and $2^{56}$. In other words, it means taking the inverse of the result of multiplying $7^{19}$ and $2^{56}$.
Since the expression cannot be simplified further without additional context or values for $7$ and $2$, we can conclude that the simplified expression is $\frac{1}{7^{19} \cdot 2^{56}}$.
bro its still wrong!!
Apologies for the confusion. Let's correct the mistake.
The expression we need to simplify is:
$(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot 4^{-13}$
To simplify this expression and eliminate the negative exponents, we can use the fact that $a^{{-n}} = \frac{1}{{a^n}}$.
Rewriting the expression, we have:
$(\frac{1}{4^3} \cdot 7^{2})^{-5}/7^9 \cdot \frac{1}{4^{13}}$
Now, let's simplify the exponents and multiply the terms:
$(\frac{1}{64} \cdot 49)^{-5}/7^9 \cdot \frac{1}{4^{13}}$
Simplifying further:
$\frac{1}{(\frac{49}{64})^5} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$
Now, let's evaluate $(\frac{49}{64})^5$:
$(\frac{49}{64})^5 = (\frac{7^2}{2^6})^5 = \frac{7^{10}}{2^{30}}$
Substituting this back into the expression:
$\frac{1}{\frac{7^{10}}{2^{30}}} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$
Next, we can simplify $4^{13}$:
$4^{13} = (2^2)^{13} = 2^{26}$
Substituting this back into the expression:
$\frac{1}{\frac{7^{10}}{2^{30}}} \cdot \frac{1}{7^9} \cdot \frac{1}{2^{26}}$
To simplify the expression, we can combine the terms with the same base:
$\frac{1}{(\frac{7^{10}}{2^{30}}) \cdot 7^9 \cdot 2^{26}}$
Using the properties of exponents, we can simplify further:
$\frac{2^{30}}{7^{10}} \cdot \frac{1}{7^9} \cdot \frac{1}{2^{26}}$
Now, let's combine the terms:
$\frac{2^{30}}{7^{10} \cdot 7^9} \cdot \frac{1}{2^{26}}$
Simplifying the exponents:
$\frac{2^{30}}{7^{10 + 9}} \cdot \frac{1}{2^{26}}$
$\frac{2^{30}}{7^{19}} \cdot \frac{1}{2^{26}}$
Applying the properties of exponents again, we can simplify to:
$\frac{2^{30 - 26}}{7^{19}}$
$\frac{2^4}{7^{19}}$
Simplifying $2^4$:
$2^4 = 16$
Therefore, the simplified expression is $\frac{16}{7^{19}}$ or $\frac{16}{7^{19}}$.
{16}{7^{19} is?
$16/7^{19}$ represents the fraction 16 divided by $7^{19}$. It means the same as $\frac{16}{7^{19}}$.
well.. ITS WRONG!
Apologies for the mistake. Let's correct it.
The expression we need to simplify is:
$(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot 4^{-13}$
To simplify this expression, we can use the properties of integer exponents.
First, let's simplify the exponents and rewrite the expression:
$4^3 = 4 \cdot 4 \cdot 4$
$7^{-2} = \frac{1}{7^2}$
$7^9$ remains as it is
$4^{-13} = \frac{1}{4^{13}}$
Now, let's substitute these values back into the expression:
$(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot