Question

find the cube roots for each number
3 square root of 45 estimate - show your reasoning

Answer 1

To find the cube roots for the number \( 3 \sqrt{45} \), we first need to simplify \( 3 \sqrt{45} \).

### Step 1: Simplify \( \sqrt{45} \)

The square root of 45 can be simplified as follows:

\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
\]

### Step 2: Multiply by 3

Now we can rewrite \( 3 \sqrt{45} \):

\[
3 \sqrt{45} = 3 \times 3 \sqrt{5} = 9 \sqrt{5}
\]

### Step 3: Estimate \( 9 \sqrt{5} \)

To estimate \( 9 \sqrt{5} \), we need to estimate \( \sqrt{5} \). The approximate value of \( \sqrt{5} \) is about 2.236.

Now, we can calculate:

\[
9 \sqrt{5} \approx 9 \times 2.236 = 20.124
\]

### Final Calculation for Cube Roots

Now we will find the cube root of \( 9\sqrt{5} \):

\[
x = (9\sqrt{5})^{1/3}
\]

To find this fractionally, we can break it down:

\[
= 9^{1/3} \cdot (\sqrt{5})^{1/3} = 9^{1/3} \cdot 5^{1/6}
\]

Next, we approximate \( 9^{1/3} \) and \( 5^{1/6} \):

- \( 9^{1/3} \approx 2.08 \) (since \( 2^3 = 8 \) is close to \( 9 \))
- \( 5^{1/6} \approx 1.43 \) (since \( 2^6 = 64 \) is much larger than \( 5 \) and \( 1.5^6 = 11.39 \) is closer)

Now multiplying these approximations to estimate the cube root:

\[
(9\sqrt{5})^{1/3} \approx 2.08 \times 1.43 \approx 2.97
\]

### Conclusion

Thus, the cube roots of \( 3 \sqrt{45} \) or \( 9 \sqrt{5} \) is approximately \( 2.97 \).

Answer 2

75