Sure, let's solve each part step-by-step.
### Part a: [tex]\(\left(2.0 \times 10^5\right)^2\)[/tex]
To solve this, we need to square both the coefficient and the exponent separately:
1. Coefficient: [tex]\(2.0^2 = 4.0\)[/tex]
2. Exponent: [tex]\((10^5)^2 = 10^{5 \times 2} = 10^{10}\)[/tex]
Combining these results, we get:
[tex]\[ \left(2.0 \times 10^5\right)^2 = 4.0 \times 10^{10} \][/tex]
So, the answer is:
[tex]\[ 4.0 \times 10^{10} = 40,000,000,000.0 \][/tex]
### Part b: [tex]\(\sqrt{9.00 \times 10^6}\)[/tex]
To solve this, we need to take the square root of both the coefficient and the exponent separately:
1. Coefficient: [tex]\(\sqrt{9.00} = 3.00\)[/tex]
2. Exponent: [tex]\(\sqrt{10^6} = 10^{6/2} = 10^3\)[/tex]
Combining these results, we get:
[tex]\[ \sqrt{9.00 \times 10^6} = 3.00 \times 10^3 \][/tex]
So, the answer is:
[tex]\[ 3.00 \times 10^3 = 3,000.0 \][/tex]
### Part c: [tex]\(\left(1.0 \times 10^6\right) \times \left(2.5 \times 10^4\right)\)[/tex]
To solve this, we need to multiply both the coefficients and the exponents:
1. Coefficient: [tex]\(1.0 \times 2.5 = 2.5\)[/tex]
2. Exponent: [tex]\(10^6 \times 10^4 = 10^{6+4} = 10^{10}\)[/tex]
Combining these results, we get:
[tex]\[ \left(1.0 \times 10^6\right) \times \left(2.5 \times 10^4\right) = 2.5 \times 10^{10} \][/tex]
So, the answer is:
[tex]\[ 2.5 \times 10^{10} = 25,000,000,000.0 \][/tex]
### Part d: [tex]\(\frac{4.0 \times 10^{-2}}{2.0 \times 10^{-5}}\)[/tex]
To solve this, we need to divide both the coefficients and the exponents:
1. Coefficient: [tex]\(\frac{4.0}{2.0} = 2.0\)[/tex]
2. Exponent: [tex]\(\frac{10^{-2}}{10^{-5}} = 10^{-2 - (-5)} = 10^{3}\)[/tex]
Combining these results, we get:
[tex]\[ \frac{4.0 \times 10^{-2}}{2.0 \times 10^{-5}} = 2.0 \times 10^{3} \][/tex]
So, the answer is:
[tex]\[ 2.0 \times 10^{3} = 2,000.0 \][/tex]
Putting it all together, the final answers are:
a. [tex]\(40,000,000,000.0\)[/tex]
b. [tex]\(3,000.0\)[/tex]
c. [tex]\(25,000,000,000.0\)[/tex]
d. [tex]\(2,000.0\)[/tex]
