must lie between the whole numbers 9 and 10 because 9 to the power of 2 = 81 and10 to the power of 2 = 100, and 97 lies between these values.

Question

to one decimal place, 97 must lie between __ and __

Answer 1

To determine the whole numbers between which 97 lies when considering the square root, we can find the square roots of 97.

The square root of 97 is approximately 9.8, since:

• \(9^2 = 81\)
• \(10^2 = 100\)

Thus, since \(9^2 < 97 < 10^2\), we can conclude:

The square root of 97 must lie between 9.8 and 9.9 (to one decimal place).

Answer 2

must lie between the whole numbers 9 and 10 because 9 to the power of 2 = 81 and10 to the power of 2 = 100, and 97 lies between these values.
to one decimal place, 97 must lie between 9.8 and 9.9 because 9.8 to the power of 2 = 96.04 and 9.9 to the power of 2 = 98.01, and 97 lives between the values.

To Two decimal places, 97 must lie between ___ and __

Answer 3

To find the values to two decimal places between which 97 lies, we can refine our approach by calculating the square roots of values slightly more precise than 9.8 and 9.9.

Let's check 9.85 and 9.86:

1. Compute \(9.85^2\): \[ 9.85^2 = 97.0225 \]

2. Compute \(9.86^2\): \[ 9.86^2 = 97.0596 \]

3. Compute \(9.84^2\): \[ 9.84^2 = 96.8064 \]

From these calculations:

• \(9.84^2 = 96.8064 < 97 < 9.85^2 = 97.0225\)

This indicates that the square root of 97 must lie between 9.84 and 9.85 to two decimal places.

Therefore, to two decimal places, 97 must lie between 9.84 and 9.85.

Answer 4

78 must lie between the whole numbers 8 and 9 because 8 to the power of 2 = 64 and 9 to the power of 2 = 81, and 78 lies between these values.

to one decimal place, 78 must lie between __ and __

Answer 5

To find the range between which the square root of 78 lies to one decimal place, we can calculate the square roots of values around 8 and 9.

1. Start with \(8.8\): \[ 8.8^2 = 77.44 \]

2. Now calculate \(8.9\): \[ 8.9^2 = 79.21 \]

Since \(77.44 < 78 < 79.21\), we see that the square root of 78 must lie between \(8.8\) and \(8.9\).

To summarize, to one decimal place, 78 must lie between 8.8 and 8.9.

Answer 6

78 must lie between the whole numbers 8 and 9 because 8 to the power of 2 = 64 and 9 to the power of 2 = 81, and 78 lies between these values.
to one decimal place, 78 must lie between 8.8 and 8.9 because 8.8 to the power of 2 = 77.44 and 8.9 to the power of 2 = 79.21, and 78 lies between these values.

To decimal places, 78 must lie between ___ and __

Answer 7

To find the range for the square root of 78 to two decimal places, we can calculate the squares of values between 8.8 and 8.9, specifically at \(8.85\) and \(8.86\):

1. Calculate \(8.85^2\): \[ 8.85^2 = 78.2225 \]

2. Calculate \(8.86^2\): \[ 8.86^2 = 78.2596 \]

3. Calculate \(8.84^2\): \[ 8.84^2 = 78.1856 \]

From these calculations, we see that:

• \(8.84^2 = 78.1856 < 78 < 8.85^2 = 78.2225\)

This indicates that the square root of 78 must lie between 8.84 and 8.85 to two decimal places.

Therefore, to two decimal places, 78 must lie between 8.84 and 8.85.

Answer 8

the answer is NOT 8.84 and 8.85

Answer 9

I apologize for the oversight. Let’s carefully calculate the square of numbers around the estimates of the square root of 78 to two decimal places.

We already know:

• \(8.8^2 = 77.44\)
• \(8.9^2 = 79.21\)

To get more precise results, let's try squares of numbers between 8.8 and 8.9, specifically \(8.85\), \(8.86\), and \(8.87\).

1. Calculate \(8.85^2\): \[ 8.85^2 = 78.2225 \]

2. Calculate \(8.86^2\): \[ 8.86^2 = 78.2596 \]

3. Calculate \(8.87^2\): \[ 8.87^2 = 78.4969 \]

Now we have:

• \(8.85^2 = 78.2225\) (this is above 78)
• \(8.86^2 = 78.2596\) (this is also above 78)
• \(8.84^2 = 78.1856\) (this is below 78)

Now calculate \(8.83^2\): \[ 8.83^2 = 78.1289 \]

At this point, we find:

• \(8.83^2 = 78.1289 < 78 < 8.84^2 = 78.1856\)

Next, let’s calculate \(8.82^2\): \[ 8.82^2 = 77.8724 \]

Now we can conclude:

• \(8.82^2 = 77.8724 < 78 < 8.83^2 = 78.1289\)

This establishes that:

To two decimal places, the square root of 78 must lie between 8.82 and 8.83.

Thank you for your patience!

Answer 10

78 must lie between the whole numbers 8 and 9 because 8 to the power of 2 = 64 and 9 to the power of 2 = 81, and 78 lies between these values.
to one decimal place, 78 must lie between 8.8 and 8.9 because 8.8 to the power of 2 = 77.44 and 8.9 to the power of 2 = 79.21, and 78 lies between these values.

To decimal places, 78 must lie between ___ and __

the asnwer is not 8.84 and 8.85 nor 8.82 and 8.83

Answer 11

Let’s re-evaluate the calculations for the square root of 78, with a focus on identifying the correct range to two decimal places.

From previous calculations, we have:

• \(8.8^2 = 77.44\)
• \(8.9^2 = 79.21\)

Next, we need to find more precise values between \(8.8\) and \(8.9\) by examining values such as \(8.85\) and \(8.86\):

1. Calculate \(8.85^2\): \[ 8.85^2 = 78.2225 \]

2. Calculate \(8.86^2\): \[ 8.86^2 = 78.2596 \]

3. Calculate \(8.84^2\): \[ 8.84^2 = 78.1856 \]

Since we know:

• \(8.84^2 = 78.1856 < 78 < 8.85^2 = 78.2225\)

Now, let’s further refine our search with values like \(8.83\):

4. Calculate \(8.83^2\): \[ 8.83^2 = 78.1289 \]

Running through the calculations again:

• \(8.83^2 = 78.1289 < 78\)
• \(8.84^2 = 78.1856 > 78\)

To find a narrower range, let’s check \(8.81\) and \(8.82\):

5. Calculate \(8.81^2\): \[ 8.81^2 = 77.6161 \]

6. Calculate \(8.82^2\): \[ 8.82^2 = 77.8724 \]

Now we see:

• \(8.81^2 = 77.6161 < 78\)
• \(8.82^2 = 77.8724 < 78\) (not helpful)

So now, let’s focus back on our calculations. Here’s a final review up to \(8.87\) and \(8.88\):

7. Calculate \(8.87^2\): \[ 8.87^2 = 78.4969 \]

8. Calculate \(8.88^2\): \[ 8.88^2 = 78.6544 \]

Final findings:

• From all these, we see:
  • \(8.84^2 = 78.1856 < 78 < 8.85^2 = 78.2225\)
  • So, our range, where \(78\) lies with precision to two decimal places, is between 8.83 and 8.84.

To summarize:

To two decimal places, \(78\) must lie between 8.83 and 8.84.

Thank you for your patience, and I hope this answer aligns with your expectations!