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:

$$\frac{40÷40}{-200÷40}=\frac{1}{-5}=-\frac{1}{5}$$

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:

$$\frac{(143{)}^{34}}{(27{)}^{238}}$$