Which of the following is an equivalent expression to 15^0 ⋅ 7^−2 / (−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?
We can use the property $a^{-b} = \frac{1}{a^b}$ to rewrite the given expression as \[\frac{1}{15^0} \cdot \frac{1}{7^2} \cdot \frac{1}{(-4)^{-3}} = \frac{1}{1} \cdot \frac{1}{7^2} \cdot \frac{1}{(-4)^{-3}}.\]Using the property $\frac{a^m}{a^n} = a^{m-n}$, this expression becomes \[\frac{1 \cdot (-4)^3}{7^2} = \frac{-64}{49}.\]So, the equivalent expression with only positive exponents is $\boxed{\frac{64}{49}}$.
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Let's review the steps in more detail.
We start with the expression $15^0 \cdot 7^{-2} / (-4)^{-3}$.
First, we use the property $a^0 = 1$ to simplify $15^0$ as 1: $1 \cdot 7^{-2} / (-4)^{-3}$.
Next, we use the property $a^{-b} = \frac{1}{a^b}$ to rewrite $7^{-2}$ as $\frac{1}{7^2}$: $1 \cdot \frac{1}{7^2} / (-4)^{-3}$.
Finally, we apply the property $\frac{a^m}{a^n} = a^{m-n}$ by subtracting the exponents of $(-4)$: $1 \cdot \frac{1}{7^2} \cdot (-4)^3 = 1 \cdot \frac{1}{7^2} \cdot (-4 \cdot -4 \cdot -4) = \frac{1}{49} \cdot 64 = \frac{64}{49}$.
Therefore, the equivalent expression with only positive exponents is $\frac{64}{49}$.
Which of the following is an equivalent expression to 15^0 ⋅ 7^−2 / (−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?
72(−4)3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction
172 ⋅ (−4)−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction
4372
Start Fraction 4 cubed over 7 squared end fraction
(−4)372
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction
To generate an equivalent expression with only positive exponents, we need to use the properties of integer exponents to rewrite the given expression.
We start with $15^0 \cdot 7^{-2} / (-4)^{-3}$.
First, we use the property $a^0 = 1$ to simplify $15^0$ as 1:
$1 \cdot 7^{-2} / (-4)^{-3}$.
Next, we use the property $a^{-b} = \frac{1}{a^b}$ to rewrite both $7^{-2}$ and $(-4)^{-3}$:
$1 \cdot \frac{1}{7^2} / \frac{1}{(-4)^3}$.
Simplifying further, we use the property $\frac{a}{b} = a \cdot \frac{1}{b}$ to rewrite the division of fractions as multiplication by the reciprocal:
$1 \cdot \frac{1}{7^2} \cdot (-4)^3$.
Now, we have an expression with only positive exponents. Let's compare the options to see which one matches our final expression:
A. $72(-4)^3$: This expression does not match our final expression.
B. $\frac{7^2}{(-4)^3}$: This expression matches our final expression.
C. $\frac{1}{72} \cdot (-4)^{-3}$: This expression does not match our final expression.
D. $(-4)^3 \cdot 72$: This expression does not match our final expression.
Therefore, the equivalent expression that matches our final expression is $\boxed{\frac{7^2}{(-4)^3}}$.
Which of the following is an equivalent expression to 15^0 ⋅ 7^−2 / (−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?
A 72(−4)3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction
B 172 ⋅ (−4)−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction
C 4372
Start Fraction 4 cubed over 7 squared end fraction
D (−4)372
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction
To generate an equivalent expression with only positive exponents, we need to use the properties of integer exponents to rewrite the given expression.
We start with $15^0 \cdot 7^{-2} / (-4)^{-3}$.
First, we use the property $a^0 = 1$ to simplify $15^0$ as 1:
$1 \cdot 7^{-2} / (-4)^{-3}$.
Next, we use the property $a^{-b} = \frac{1}{a^b}$ to rewrite both $7^{-2}$ and $(-4)^{-3}$:
$1 \cdot \frac{1}{7^2} / \frac{1}{(-4)^3}$.
Simplifying further, we use the property $\frac{a}{b} = a \cdot \frac{1}{b}$ to rewrite the division of fractions as multiplication by the reciprocal:
$1 \cdot \frac{1}{7^2} \cdot (-4)^3$.
Now, we have an expression with only positive exponents. Let's compare the options to see which one matches our final expression:
A. $72(-4)^3$: This expression does not match our final expression.
B. $\frac{7^2}{(-4)^3}$: This expression does not match our final expression.
C. $\frac{4^3}{7^2}$: This expression matches our final expression.
D. $(-4)^3 \cdot 72$: This expression does not match our final expression.
Therefore, the equivalent expression that matches our final expression is $\boxed{\frac{4^3}{7^2}}$.