Question
Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7 x 3^−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
Answers
Answered by
GPT 3.5
To apply the properties of integer exponents, we can use the property $a^m \cdot a^n = a^{m+n}$.
Using this property, we can rewrite $3^7 \cdot 3^{-9}$ as $3^{7 - 9}$.
Simplifying further, $3^{7 - 9} = 3^{-2}$.
To get rid of the negative exponent, we use the property $a^{-n} = \frac{1}{a^n}$.
So, $3^{-2} = \frac{1}{3^2}$.
Finally, we simplify $\frac{1}{3^2}$ to get the answer: $\boxed{\frac{1}{9}}$.
Using this property, we can rewrite $3^7 \cdot 3^{-9}$ as $3^{7 - 9}$.
Simplifying further, $3^{7 - 9} = 3^{-2}$.
To get rid of the negative exponent, we use the property $a^{-n} = \frac{1}{a^n}$.
So, $3^{-2} = \frac{1}{3^2}$.
Finally, we simplify $\frac{1}{3^2}$ to get the answer: $\boxed{\frac{1}{9}}$.
Answer
what's the answer
Answered by
GPT 3.5
The answer is $\frac{1}{9}$.
Answer
Apply the Properties of Integer Exponents to generate equivalent expressions to 2^2/2^−4 with only positive exponents, then solve the expression. Provide the solution as a whole number.
Answered by
GPT 3.5
To apply the properties of integer exponents, we can use the property $\frac{a^m}{a^n} = a^{m - n}$.
Using this property, we can rewrite $\frac{2^2}{2^{-4}}$ as $2^{2 - (-4)}$.
Simplifying further, $2^{2 - (-4)} = 2^{2 + 4}$.
Using the property $a^m \cdot a^n = a^{m+n}$, we can rewrite $2^{2 + 4}$ as $2^6$.
Therefore, the value of $2^2/2^{-4}$ is $2^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = \boxed{64}$.
Using this property, we can rewrite $\frac{2^2}{2^{-4}}$ as $2^{2 - (-4)}$.
Simplifying further, $2^{2 - (-4)} = 2^{2 + 4}$.
Using the property $a^m \cdot a^n = a^{m+n}$, we can rewrite $2^{2 + 4}$ as $2^6$.
Therefore, the value of $2^2/2^{-4}$ is $2^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = \boxed{64}$.
Answer
Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
Answered by
GPT 3.5
To apply the properties of integer exponents, we can use the property $a^0 = 1$ for any non-zero value of $a$.
Using this property, we can rewrite $4^0$ as $1$.
Similarly, we can use the property $a^{-n} = \frac{1}{a^n}$ to rewrite $5^{-2}$ as $\frac{1}{5^2}$.
For $(-2)^{-3}$, we can use the property $a^{-n} = \frac{1}{a^n}$ to rewrite $(-2)^{-3}$ as $\frac{1}{(-2)^3}$.
Now, let's simplify the expression.
$4^0 \cdot 5^{-2} \div (-2)^{-3}$ can be rewritten as $1 \cdot \frac{1}{5^2} \div \frac{1}{(-2)^3}$.
Using the property $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$, we can simplify further:
$1 \cdot \frac{1}{5^2} \div \frac{1}{(-2)^3} = 1 \cdot \frac{1}{5^2} \cdot \frac{(-2)^3}{1}$.
Now, let's evaluate each term:
$1 = 1$,
$5^2 = 5 \cdot 5 = 25$,
$(-2)^3 = -2 \cdot -2 \cdot -2 = -8$.
Therefore, the expression simplifies to $1 \cdot \frac{1}{25} \cdot \frac{-8}{1}$.
Simplifying further, $1 \cdot \frac{1}{25} \cdot \frac{-8}{1} = \frac{-8}{25}$.
Thus, the answer is $\boxed{\frac{-8}{25}}$.
Using this property, we can rewrite $4^0$ as $1$.
Similarly, we can use the property $a^{-n} = \frac{1}{a^n}$ to rewrite $5^{-2}$ as $\frac{1}{5^2}$.
For $(-2)^{-3}$, we can use the property $a^{-n} = \frac{1}{a^n}$ to rewrite $(-2)^{-3}$ as $\frac{1}{(-2)^3}$.
Now, let's simplify the expression.
$4^0 \cdot 5^{-2} \div (-2)^{-3}$ can be rewritten as $1 \cdot \frac{1}{5^2} \div \frac{1}{(-2)^3}$.
Using the property $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$, we can simplify further:
$1 \cdot \frac{1}{5^2} \div \frac{1}{(-2)^3} = 1 \cdot \frac{1}{5^2} \cdot \frac{(-2)^3}{1}$.
Now, let's evaluate each term:
$1 = 1$,
$5^2 = 5 \cdot 5 = 25$,
$(-2)^3 = -2 \cdot -2 \cdot -2 = -8$.
Therefore, the expression simplifies to $1 \cdot \frac{1}{25} \cdot \frac{-8}{1}$.
Simplifying further, $1 \cdot \frac{1}{25} \cdot \frac{-8}{1} = \frac{-8}{25}$.
Thus, the answer is $\boxed{\frac{-8}{25}}$.
Answer
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)
Answered by
GPT 3.5
To apply the properties of integer exponents, we can use the property $(a^m)^n = a^{m \cdot n}$.
Using this property, we can rewrite $(14^3 \cdot 27^{-7})^0$ as $1$ since any non-zero number raised to the power of 0 is equal to 1.
Now, we have $1/3^4$ left in the expression.
Using the property $a^{-n} = \frac{1}{a^n}$, we can rewrite $3^{-4}$ as $\frac{1}{3^4}$.
Therefore, the expression $(14^3 \cdot 27^{-7})^0/3^4$ can be simplified to $1/3^4$.
Simplifying further, $1/3^4 = 1/3 \cdot 3 \cdot 3 \cdot 3 = 1/(3 \cdot 3 \cdot 3 \cdot 3) = \boxed{\frac{1}{81}}$
Using this property, we can rewrite $(14^3 \cdot 27^{-7})^0$ as $1$ since any non-zero number raised to the power of 0 is equal to 1.
Now, we have $1/3^4$ left in the expression.
Using the property $a^{-n} = \frac{1}{a^n}$, we can rewrite $3^{-4}$ as $\frac{1}{3^4}$.
Therefore, the expression $(14^3 \cdot 27^{-7})^0/3^4$ can be simplified to $1/3^4$.
Simplifying further, $1/3^4 = 1/3 \cdot 3 \cdot 3 \cdot 3 = 1/(3 \cdot 3 \cdot 3 \cdot 3) = \boxed{\frac{1}{81}}$
Answer
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.(1 point)
Answered by
GPT 3.5
To apply the properties of integer exponents, we can use the property $(a^m)^n = a^{m \cdot n}$.
Using this property, we can rewrite $(4^3 \cdot 7^{-2})^{-5}$ as $(4^3)^{-5} \cdot (7^{-2})^{-5}$.
Simplifying further, $(4^3)^{-5} = 4^{3 \cdot -5}$.
Again, using the property $a^{-n} = \frac{1}{a^n}$, we can rewrite $(7^{-2})^{-5}$ as $\frac{1}{(7^{-2})^5}$.
Let's simplify each term:
$4^{3 \cdot -5} = 4^{-15}$.
Using the property $a^m \cdot a^n = a^{m+n}$, we can rewrite $4^{-15}$ as $1/4^{15}$.
Similarly, $(7^{-2})^5$ can be computed as $7^{-2 \cdot 5} = 7^{-10}$.
Thus, the expression $(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot 4^{-13}$ can be simplified to $(1/4^{15})/7^9 \cdot 4^{-13}$.
To divide by $7^9$, we use the property $\frac{a}{b} = a \cdot \frac{1}{b}$:
$(1/4^{15})/7^9 \cdot 4^{-13} = (1/4^{15}) \cdot \frac{1}{7^9} \cdot 4^{-13}$.
Now, let's simplify each term:
$1/4^{15} = 1/4^{15}$,
$1/7^9 = \frac{1}{7^9}$,
$4^{-13} = \frac{1}{4^{13}}$.
Putting it all together, we have:
$(1/4^{15}) \cdot \frac{1}{7^9} \cdot 4^{-13} = \frac{1}{4^{15}} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$.
Using the property $\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$, we can combine these fractions into one:
$\frac{1}{4^{15}} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}} = \frac{1 \cdot 1 \cdot 1}{4^{15} \cdot 7^9 \cdot 4^{13}}$.
Simplifying the denominator:
$4^{15} \cdot 7^9 \cdot 4^{13} = 4^{15 + 13} \cdot 7^9$.
Using the property $a^m \cdot a^n = a^{m+n}$:
$4^{15 + 13} \cdot 7^9 = 4^{28} \cdot 7^9$.
Finally, the expression simplifies to:
$\frac{1}{4^{15} \cdot 7^9 \cdot 4^{13}} = \frac{1}{4^{28} \cdot 7^9} = \boxed{\frac{1}{4^{28} \cdot 7^9}}$.
Using this property, we can rewrite $(4^3 \cdot 7^{-2})^{-5}$ as $(4^3)^{-5} \cdot (7^{-2})^{-5}$.
Simplifying further, $(4^3)^{-5} = 4^{3 \cdot -5}$.
Again, using the property $a^{-n} = \frac{1}{a^n}$, we can rewrite $(7^{-2})^{-5}$ as $\frac{1}{(7^{-2})^5}$.
Let's simplify each term:
$4^{3 \cdot -5} = 4^{-15}$.
Using the property $a^m \cdot a^n = a^{m+n}$, we can rewrite $4^{-15}$ as $1/4^{15}$.
Similarly, $(7^{-2})^5$ can be computed as $7^{-2 \cdot 5} = 7^{-10}$.
Thus, the expression $(4^3 \cdot 7^{-2})^{-5}/7^9 \cdot 4^{-13}$ can be simplified to $(1/4^{15})/7^9 \cdot 4^{-13}$.
To divide by $7^9$, we use the property $\frac{a}{b} = a \cdot \frac{1}{b}$:
$(1/4^{15})/7^9 \cdot 4^{-13} = (1/4^{15}) \cdot \frac{1}{7^9} \cdot 4^{-13}$.
Now, let's simplify each term:
$1/4^{15} = 1/4^{15}$,
$1/7^9 = \frac{1}{7^9}$,
$4^{-13} = \frac{1}{4^{13}}$.
Putting it all together, we have:
$(1/4^{15}) \cdot \frac{1}{7^9} \cdot 4^{-13} = \frac{1}{4^{15}} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}}$.
Using the property $\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$, we can combine these fractions into one:
$\frac{1}{4^{15}} \cdot \frac{1}{7^9} \cdot \frac{1}{4^{13}} = \frac{1 \cdot 1 \cdot 1}{4^{15} \cdot 7^9 \cdot 4^{13}}$.
Simplifying the denominator:
$4^{15} \cdot 7^9 \cdot 4^{13} = 4^{15 + 13} \cdot 7^9$.
Using the property $a^m \cdot a^n = a^{m+n}$:
$4^{15 + 13} \cdot 7^9 = 4^{28} \cdot 7^9$.
Finally, the expression simplifies to:
$\frac{1}{4^{15} \cdot 7^9 \cdot 4^{13}} = \frac{1}{4^{28} \cdot 7^9} = \boxed{\frac{1}{4^{28} \cdot 7^9}}$.