Answer :
Let's estimate the given problems step-by-step:
### a) [tex]\( 78.45 + 51.02 \)[/tex]
First, round each number to a whole number:
- [tex]\( 78.45 \approx 78 \)[/tex]
- [tex]\( 51.02 \approx 51 \)[/tex]
Add these rounded numbers:
[tex]\[ 78 + 51 = 129 \][/tex]
So, we estimate [tex]\( 78.45 + 51.02 \approx 129 \)[/tex].
### b) [tex]\( 168.3 - 87.09 \)[/tex]
First, round each number to a whole number:
- [tex]\( 168.3 \approx 168 \)[/tex]
- [tex]\( 87.09 \approx 87 \)[/tex]
Subtract these rounded numbers:
[tex]\[ 168 - 87 = 81 \][/tex]
So, we estimate [tex]\( 168.3 - 87.09 \approx 81 \)[/tex].
### c) [tex]\( 2.93 \times 3.14 \)[/tex]
First, round each number:
- [tex]\( 2.93 \approx 3 \)[/tex]
- [tex]\( 3.14 \approx 3 \)[/tex]
Multiply these rounded numbers:
[tex]\[ 3 \times 3 = 9 \][/tex]
So, we estimate [tex]\( 2.93 \times 3.14 \approx 9 \)[/tex].
### d) [tex]\( 84.2 \div 19.5 \)[/tex]
First, round each number:
- [tex]\( 84.2 \approx 84 \)[/tex]
- [tex]\( 19.5 \approx 20 \)[/tex]
Divide these rounded numbers:
[tex]\[ 84 \div 20 = 4.2 \][/tex]
So, we estimate [tex]\( 84.2 \div 19.5 \approx 4.2 \)[/tex].
### e) [tex]\( \frac{4.3 \times 752}{15.6} \)[/tex]
First, approximate the product in the numerator:
- [tex]\( 4.3 \approx 4 \)[/tex]
- [tex]\( 752 \approx 750 \)[/tex]
Multiply the rounded numbers:
[tex]\[ 4 \times 750 = 3000 \][/tex]
Then, round the denominator:
- [tex]\( 15.6 \approx 16 \)[/tex]
Divide the resulting numbers:
[tex]\[ \frac{3000}{16} \approx 187.5 \][/tex]
So, we estimate [tex]\( \frac{4.3 \times 752}{15.6} \approx 187.5 \)[/tex].
### f) [tex]\( \frac{(9.8)^3}{(2.2)^2} \)[/tex]
First, approximate the cube and the square:
- [tex]\( 9.8 \approx 10 \)[/tex]
- [tex]\( 2.2 \approx 2 \)[/tex]
Cube and square the rounded numbers:
[tex]\[ 10^3 = 1000 \][/tex]
[tex]\[ 2^2 = 4 \][/tex]
Divide these numbers:
[tex]\[ \frac{1000}{4} = 250 \][/tex]
So, we estimate [tex]\( \frac{(9.8)^3}{(2.2)^2} \approx 250 \)[/tex].
### g) [tex]\( \frac{\sqrt[3]{78} \times 6}{5^3} \)[/tex]
First, approximate the cube root and the numbers in the fraction:
- [tex]\( 78 \approx 80 \)[/tex]
- [tex]\( \sqrt[3]{80} \approx 4.3 \)[/tex] (a rough estimate)
Multiply by 6:
[tex]\[ 4.3 \times 6 \approx 25.8 \][/tex]
Then, cube the denominator:
[tex]\[ 5^3 = 125 \][/tex]
Divide the estimated numbers:
[tex]\[ \frac{25.8}{125} \approx 0.206 \][/tex]
So, we estimate [tex]\( \frac{\sqrt[3]{78} \times 6}{5^3} \approx 0.206 \)[/tex].
### h) [tex]\( \frac{38 \times 6^3}{\sqrt[4]{9900}} \)[/tex]
First, approximate the cube, fourth root, and intermediate results:
- [tex]\( 6^3 = 216 \)[/tex]
- [tex]\( 38 \times 216 = 8208 \)[/tex] (approximately)
- [tex]\( \sqrt[4]{9900} \approx 10 \)[/tex] (considering it's close to [tex]\( \sqrt[4]{10000} = 10 \)[/tex])
Divide:
[tex]\[ \frac{8208}{10} \approx 821 \][/tex]
So, we estimate [tex]\( \frac{38 \times 6^3}{\sqrt[4]{9900}} \approx 821 \)[/tex].
### i) [tex]\( \sqrt[4]{17} \times \sqrt[4]{15} \)[/tex]
Estimate the fourth roots:
- [tex]\( \sqrt[4]{16} = 2 \)[/tex] (as 16 is close to 17 and 15)
- Both [tex]\( \sqrt[4]{17} \approx 2 \)[/tex] and [tex]\( \sqrt[4]{15} \approx 2 \)[/tex]
Multiply the estimated roots:
[tex]\[ 2 \times 2 = 4 \][/tex]
So, we estimate [tex]\( \sqrt[4]{17} \times \sqrt[4]{15} \approx 4 \)[/tex].
This completes our step-by-step estimation for each problem.
### a) [tex]\( 78.45 + 51.02 \)[/tex]
First, round each number to a whole number:
- [tex]\( 78.45 \approx 78 \)[/tex]
- [tex]\( 51.02 \approx 51 \)[/tex]
Add these rounded numbers:
[tex]\[ 78 + 51 = 129 \][/tex]
So, we estimate [tex]\( 78.45 + 51.02 \approx 129 \)[/tex].
### b) [tex]\( 168.3 - 87.09 \)[/tex]
First, round each number to a whole number:
- [tex]\( 168.3 \approx 168 \)[/tex]
- [tex]\( 87.09 \approx 87 \)[/tex]
Subtract these rounded numbers:
[tex]\[ 168 - 87 = 81 \][/tex]
So, we estimate [tex]\( 168.3 - 87.09 \approx 81 \)[/tex].
### c) [tex]\( 2.93 \times 3.14 \)[/tex]
First, round each number:
- [tex]\( 2.93 \approx 3 \)[/tex]
- [tex]\( 3.14 \approx 3 \)[/tex]
Multiply these rounded numbers:
[tex]\[ 3 \times 3 = 9 \][/tex]
So, we estimate [tex]\( 2.93 \times 3.14 \approx 9 \)[/tex].
### d) [tex]\( 84.2 \div 19.5 \)[/tex]
First, round each number:
- [tex]\( 84.2 \approx 84 \)[/tex]
- [tex]\( 19.5 \approx 20 \)[/tex]
Divide these rounded numbers:
[tex]\[ 84 \div 20 = 4.2 \][/tex]
So, we estimate [tex]\( 84.2 \div 19.5 \approx 4.2 \)[/tex].
### e) [tex]\( \frac{4.3 \times 752}{15.6} \)[/tex]
First, approximate the product in the numerator:
- [tex]\( 4.3 \approx 4 \)[/tex]
- [tex]\( 752 \approx 750 \)[/tex]
Multiply the rounded numbers:
[tex]\[ 4 \times 750 = 3000 \][/tex]
Then, round the denominator:
- [tex]\( 15.6 \approx 16 \)[/tex]
Divide the resulting numbers:
[tex]\[ \frac{3000}{16} \approx 187.5 \][/tex]
So, we estimate [tex]\( \frac{4.3 \times 752}{15.6} \approx 187.5 \)[/tex].
### f) [tex]\( \frac{(9.8)^3}{(2.2)^2} \)[/tex]
First, approximate the cube and the square:
- [tex]\( 9.8 \approx 10 \)[/tex]
- [tex]\( 2.2 \approx 2 \)[/tex]
Cube and square the rounded numbers:
[tex]\[ 10^3 = 1000 \][/tex]
[tex]\[ 2^2 = 4 \][/tex]
Divide these numbers:
[tex]\[ \frac{1000}{4} = 250 \][/tex]
So, we estimate [tex]\( \frac{(9.8)^3}{(2.2)^2} \approx 250 \)[/tex].
### g) [tex]\( \frac{\sqrt[3]{78} \times 6}{5^3} \)[/tex]
First, approximate the cube root and the numbers in the fraction:
- [tex]\( 78 \approx 80 \)[/tex]
- [tex]\( \sqrt[3]{80} \approx 4.3 \)[/tex] (a rough estimate)
Multiply by 6:
[tex]\[ 4.3 \times 6 \approx 25.8 \][/tex]
Then, cube the denominator:
[tex]\[ 5^3 = 125 \][/tex]
Divide the estimated numbers:
[tex]\[ \frac{25.8}{125} \approx 0.206 \][/tex]
So, we estimate [tex]\( \frac{\sqrt[3]{78} \times 6}{5^3} \approx 0.206 \)[/tex].
### h) [tex]\( \frac{38 \times 6^3}{\sqrt[4]{9900}} \)[/tex]
First, approximate the cube, fourth root, and intermediate results:
- [tex]\( 6^3 = 216 \)[/tex]
- [tex]\( 38 \times 216 = 8208 \)[/tex] (approximately)
- [tex]\( \sqrt[4]{9900} \approx 10 \)[/tex] (considering it's close to [tex]\( \sqrt[4]{10000} = 10 \)[/tex])
Divide:
[tex]\[ \frac{8208}{10} \approx 821 \][/tex]
So, we estimate [tex]\( \frac{38 \times 6^3}{\sqrt[4]{9900}} \approx 821 \)[/tex].
### i) [tex]\( \sqrt[4]{17} \times \sqrt[4]{15} \)[/tex]
Estimate the fourth roots:
- [tex]\( \sqrt[4]{16} = 2 \)[/tex] (as 16 is close to 17 and 15)
- Both [tex]\( \sqrt[4]{17} \approx 2 \)[/tex] and [tex]\( \sqrt[4]{15} \approx 2 \)[/tex]
Multiply the estimated roots:
[tex]\[ 2 \times 2 = 4 \][/tex]
So, we estimate [tex]\( \sqrt[4]{17} \times \sqrt[4]{15} \approx 4 \)[/tex].
This completes our step-by-step estimation for each problem.
