Apply the Properties of Integer Exponents to generate equivalent expressions to 40 ⋅ 5−2(−2)−3

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

Answer 1

To simplify the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) using the properties of exponents, let's first rewrite the negative exponents as positive exponents.

1. Rewrite \( 5^{-2} \) as \( \frac{1}{5^2} \).
2. Rewrite \( (-2)^{-3} \) as \( \frac{1}{(-2)^3} \).

Now we can rewrite the expression:

\[ 40 \cdot 5^{-2} \cdot (-2)^{-3} = 40 \cdot \frac{1}{5^2} \cdot \frac{1}{(-2)^3} \]

3. Calculate \( 5^2 \) and \( (-2)^3 \):

\[ 5^2 = 25 \] \[ (-2)^3 = -8 \]

4. Substitute these values back into the expression:

\[ 40 \cdot \frac{1}{25} \cdot \frac{1}{-8} = \frac{40}{25 \cdot -8} \]

5. Calculate \( 25 \cdot -8 \):

\[ 25 \cdot -8 = -200 \]

6. So now our expression looks like:

\[ \frac{40}{-200} \]

7. Simplify the fraction:

\[ \frac{40 \div 40}{-200 \div 40} = \frac{1}{-5} = -\frac{1}{5} \]

Thus, the simplified fraction of the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) is

\[ -\frac{1}{5} \]

Answer 2

Apply the Properties of Integer Exponents to generate equivalent expressions to (143⋅27−7)034
with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents. (1 point)

Answer 3

To simplify the expression \((143 \cdot 27^{-7})^{034}\) using the properties of integer exponents, we will first rewrite any negative exponents as positive exponents and simplify the expression step by step.

Step 1: Rewrite the expression

We can notice that \(034\) is simply \(34\) as leading zeros do not affect the numerical value. So we rewrite it as:

\[ (143 \cdot 27^{-7})^{34} \]

Step 2: Apply exponent multiplication

Now we can apply the distributive property of exponents:

\[ (143)^{34} \cdot (27^{-7})^{34} \]

Using the property \( (a^m) \cdot (b^n) = a^m \cdot b^n \):

\[ = (143)^{34} \cdot (27^{34})^{-7} \]

Now, apply the exponent to \(27^{-7}\):

\[ = (143)^{34} \cdot \frac{1}{(27^{7})^{34}} \]

Step 3: Simplify using properties of exponents

Using the property of exponents \((a^m)^n = a^{m \cdot n}\):

\[ (27^{7})^{34} = 27^{238} \]

So we have:

\[ = \frac{(143)^{34}}{27^{238}} \]

Step 4: Final expression

The final expression, with only positive exponents, is:

\[ \frac{(143)^{34}}{27^{238}} \]

This is already a simplified fraction with no remaining negative exponents.

Conclusion

The result is:

\[ \frac{(143)^{34}}{(27)^{238}} \]

This is the expression in its simplified fraction form. If you want numerical values for \(143^{34}\) or \(27^{238}\), they are extremely large, and practical computation may not be feasible without specialized software or tools. Hence, the answer in terms of the powers is sufficient:

\[ \frac{(143)^{34}}{(27)^{238}} \]