5 answers
To apply the Property of Negative Integer Exponents, we use the rule that states:
\[ a^{-n} = \frac{1}{a^n} \]
Using this property for \(3^{-4}\):
\[ 3^{-4} = \frac{1}{3^4} \]
Now, we need to calculate \(3^4\):
\[ 3^4 = 3 \times 3 \times 3 \times 3 \]
Calculating this step-by-step:
\[ 3 \times 3 = 9 \] \[ 9 \times 3 = 27 \] \[ 27 \times 3 = 81 \]
So, \(3^4 = 81\).
Now we substitute back:
\[ 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \]
Therefore, the final answer is:
\[ \frac{1}{81} \]
To apply the Property of Negative Integer Exponents, we first simplify the expression \(7^{-3} \times 7^{-5}\) using the following property of exponents:
\[ a^m \times a^n = a^{m+n} \]
So, we can add the exponents:
\[ 7^{-3} \times 7^{-5} = 7^{-3 + (-5)} = 7^{-8} \]
Next, we apply the property of negative exponents, which states:
\[ a^{-n} = \frac{1}{a^n} \]
Thus, we can rewrite \(7^{-8}\):
\[ 7^{-8} = \frac{1}{7^8} \]
Now, we have the expression \(7^{-3} \times 7^{-5}\) written with positive exponents only:
\[ \frac{1}{7^8} \]
Therefore, the final answer is:
\[ \frac{1}{7^8} \]
To simplify the expression \( \frac{5^{-9}}{4^{-12}} \) and express it with positive exponents only, we can use the properties of negative exponents.
First, we rewrite the negative exponents:
For \( 5^{-9} \), we can use the property \( a^{-n} = \frac{1}{a^n} \): \[ 5^{-9} = \frac{1}{5^9} \]
For \( 4^{-12} \), we apply the same property: \[ 4^{-12} = \frac{1}{4^{12}} \]
Now substituting these into the original expression:
\[ \frac{5^{-9}}{4^{-12}} = \frac{\frac{1}{5^9}}{\frac{1}{4^{12}}} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ = \frac{1}{5^9} \times 4^{12} = \frac{4^{12}}{5^{9}} \]
Now we have the expression \( \frac{4^{12}}{5^{9}} \) which contains only positive exponents.
Thus, the final answer is:
\[ \frac{4^{12}}{5^{9}} \]
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