Which of the following is the equivalent expression to (15−3 ⋅ 47)04−3 that has been generated by applying the Properties of Integer Exponents?(1 point)

Question

Responses

64
64

1⋅4−3
1 times 4 negative cubed

604−3
Start Fraction 60 over 4 negative cubed end fraction

164
Start Fraction 1 over 64 end fraction

Answer 1

To simplify the expression \( (15 - 3 \cdot 47) \cdot 4^{-3} \), we first need to calculate \( 15 - 3 \cdot 47 \).

1. Calculate \( 3 \cdot 47 \): \[ 3 \cdot 47 = 141 \]

2. Now compute \( 15 - 141 \): \[ 15 - 141 = -126 \]

3. Substitute this back into the expression: \[ -126 \cdot 4^{-3} \]

Now apply the property of exponents, \( a^{-n} = \frac{1}{a^n} \): \[ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \]

Thus: \[ -126 \cdot 4^{-3} = -126 \cdot \frac{1}{64} = \frac{-126}{64} \]

Now, we simplify \(-126/64\): \[ \frac{-126}{64} = -\frac{63}{32} \] (though this ratio does not appear in your options, we continue looking for equivalent expressions).

None of the options provided directly match our result. However, let's analyze the options in full context:

1. 64 - Not equivalent.
2. 1 ⋅ 4^{-3} - This is equal to \(4^{-3}\), which is \(\frac{1}{64}\), still not resulting in matching.
3. 60 \cdot 4^{-3} - Not equal to our simplified equivalent.
4. \(\frac{1}{64}\) - This option corresponds directly to the \(4^{-3}\) value but not to the entire expression.

Given that none represent the original calculation. Therefore, more exploration is needed to assess the options against equivalence and exponent property applications. But from the data, it appears:

The best answer aligning closely to fractional equivalence is 164 (Start Fraction 1 over 64 end Fraction), as it reflects the value of \(4^{-3}\) in isolation.

Thus: Answer: \(\frac{1}{64}\).

Answer 2

Which of the following is an equivalent expression to (2−3 ⋅93)−429 ⋅9−10 with only positive exponents, generated by applying the Properties of Integer Exponents?(1 point)
Responses

2−7⋅9−129⋅9−10
Start fraction 2 superscript negative 7 baseline times 9 superscript negative 1 baseline over 2 superscript 9 baseline times 9 superscript negative 10 baseline end fraction

23 ⋅92
2 cubed times 9 squared

2392
Start Fraction 2 cubed over 9 squared end fraction

212⋅9−1229⋅9−10

Answer 3

To simplify the expression \((2^{-3} \cdot 9^3)^{-429} \cdot 9^{-10}\) while ensuring all exponents are positive, we will first work through the exponent properties and then simplify.

Step 1: Apply the power of a product property

Let's first rewrite the expression:

\[ (2^{-3} \cdot 9^3)^{-429} = (2^{-3})^{-429} \cdot (9^3)^{-429} \]

Applying the property \((a^m)^n = a^{m \cdot n}\):

\[ (2^{-3})^{-429} = 2^{-3 \cdot (-429)} = 2^{1287} \] \[ (9^3)^{-429} = 9^{3 \cdot (-429)} = 9^{-1287} \]

Putting it together:

\[ (2^{-3} \cdot 9^3)^{-429} = 2^{1287} \cdot 9^{-1287} \]

Step 2: Combine with \(9^{-10}\)

Now we multiply this expression by \(9^{-10}\):

\[ 2^{1287} \cdot 9^{-1287} \cdot 9^{-10} = 2^{1287} \cdot 9^{-1287 - 10} = 2^{1287} \cdot 9^{-1297} \]

Step 3: Rewrite with positive exponents

Now we need to write \(9^{-1297}\) using positive exponents. Recall that \(a^{-n} = \frac{1}{a^n}\):

\[ 9^{-1297} = \frac{1}{9^{1297}} \]

Thus we can write:

\[ 2^{1287} \cdot 9^{-1297} = \frac{2^{1287}}{9^{1297}} \]

Step 4: Evaluate the response options

Now let's analyze the response options given the simplification:

1. \(2^{-7} \cdot 9^{-1} / (2^9 \cdot 9^{-10})\) - Not equivalent.

2. \(2^3 \cdot 9^2\) - Not equivalent.

3. \(\frac{2^3}{9^2}\) - Not equivalent.

4. \(2^{12} \cdot 9^{-12} / (2^9 \cdot 9^{-10})\) - This option would need more breakdown, but appears complex.

None of these directly match our final result of \(\frac{2^{1287}}{9^{1297}}\).

Summary

The closest I can determine is to provide \(2^{1287}/9^{1297}\) as the final form with positive exponents, but since none match the options immediately, consider reevaluating the intended outcomes of those options amidst the principles.

None of the provided options yield the completely simplified form equivalently with only positive exponents as expected from the original calculation. Thus the process confirms direct verification against response choices may be necessary.

If needing absolute instruction, none correctly align in form to \( \frac{2^{1287}}{9^{1297}}\).