Simplify this expression using the Quotient Rule of Integer Exponents: 1253÷753 .(1 point)

Question

Responses

5106
5 superscript 106 baseline

553
5 superscript 53 baseline

12106
12 superscript 106 baseline

1253÷753

Answer 1

To simplify the expression \( 125^3 \div 75^3 \) using the Quotient Rule of Integer Exponents, we can first rewrite the bases in terms of their prime factors:

1. Rewrite \( 125 \) and \( 75 \):
   • \( 125 = 5^3 \)
   • \( 75 = 3 \times 5^2 \)

Thus, we can express the entire expression as: \[ \frac{125^3}{75^3} = \frac{(5^3)^3}{(3 \times 5^2)^3} \]

2. Apply the power of a power rule: \[ = \frac{5^{3 \cdot 3}}{(3^3) \cdot (5^2)^3} = \frac{5^9}{27 \cdot 5^6} \]

3. Now we can use the Quotient Rule of Integer Exponents, which states \( \frac{a^m}{a^n} = a^{m-n} \): \[ = \frac{5^9}{27} \cdot \frac{1}{5^6} = \frac{5^{9-6}}{27} = \frac{5^3}{27} \]

Thus, the expression simplifies to \( \frac{5^3}{27} \).

None of the provided options appear to match this simplified form correctly. However, if the question intended a different interpretation, please take note. In a strict sense of simplification and re-evaluation of the options, the expression remains \( \frac{5^3}{27} \).

Answer 2

what answer is it

Answer 3

Let's clarify the given expression \( 125^3 \div 75^3 \) again and find the correct simplified answer.

1. Rewrite the bases:

   • \( 125 = 5^3 \)
   • \( 75 = 3 \times 5^2 \)

2. Rewrite the expression: \[ \frac{125^3}{75^3} = \frac{(5^3)^3}{(3 \times 5^2)^3} \]

3. Apply the power of a power: \[ = \frac{5^{9}}{(3^3) \cdot (5^2)^3} = \frac{5^9}{27 \cdot 5^6} \]

4. Use the Quotient Rule: \[ = \frac{5^{9-6}}{27} = \frac{5^3}{27} \]

Now, let's evaluate \( 5^3 \): \[ 5^3 = 125 \]

So, we have: \[ \frac{5^3}{27} = \frac{125}{27} \]

Looking back at the provided options:

1. \( 5106 \)
2. \( 553 \)
3. \( 12106 \)
4. \( 125^3 \div 75^3 \)

While none of these options represent \( \frac{5^3}{27} \) or \( \frac{125}{27} \), the calculation indicates that this is the correct simplified form based on the original operation.

If I have to pick from the options listed based solely on what seems like a typographical error or misunderstanding, option 4 is the closest because it represents the unsimplified form of the expression \( 125^3 \div 75^3\).

For the clarity of this simplification: the correct answer is \(\frac{5^3}{27}\), but if you're required to select one of the options, then choose option 4.