Question

Use the following equation to answer the questions below: ( 3.3 × 10^3 ) x/ 1.32 × 10 ^− 1 = 5.0 × 10 ^− 2 / 4.0 × 10 7

a. Explain what negative exponents mean and how they are used in relation to scientific notation.

b. Explain what steps you would use to simplify the equation.

c. Solve for x ; state your answer in scientific notation and also in decimal notation. Show your work.

d. Explain how to convert between the decimal value of x and the scientific notation value of x .

1. a. Negative exponents in scientific notation represent the number of times a number is divided by 10. For example, in the equation (3.3 × 10^3) x/ (1.32 × 10 ^− 1), the exponent -1 on the 10 in the denominator represents one division by 10, so the value of 1.32 × 10 ^− 1 is equal to 0.132. Negative exponents are used in scientific notation to represent very small numbers, since it can be more convenient to write the number as a decimal multiplied by a power of 10 rather than writing out the full decimal representation.
b. To simplify the equation, you can start by applying the property of exponents that states that x^a * x^b = x^(a+b) to the right side of the equation. This allows you to combine the two exponents of 10, so the right side of the equation becomes (5.0 × 10 ^− 2 / 4.0 × 10 ^7) = (5.0 / 4.0) * (10 ^− 2 / 10 ^7) = 1.25 * 10 ^− 9.
Next, you can apply the property of exponents that states that x^a / x^b = x^(a-b). This allows you to simplify the left side of the equation to (3.3 × 10^3) x/ (1.32 × 10 ^− 1) = (3.3 × 10^3) / (1.32 × 10 ^− 1) * x = 2.50 * 10 ^3 * x.
Finally, you can set the left side of the equation equal to the right side and solve for x.
c. 2.50 * 10 ^3 * x = 1.25 * 10 ^− 9
x = 1.25 * 10 ^− 9 / 2.50 * 10 ^3
x = 5.00 * 10 ^− 12
The value of x in decimal notation is 5.00 * 10 ^− 12, which is equal to 0.000000000005.
d. To convert between the decimal value of x and the scientific notation value of x, you can use the following steps:
1. To convert from decimal to scientific notation, start by moving the decimal point in the decimal number until there is only one non-zero digit to the left of the decimal point. For example, to convert 0.000000000005 to scientific notation, you would move the decimal point 12 places to the left, so the number becomes 5.00 * 10 ^− 12. The exponent on the 10 represents the number of places the decimal point was moved.
2. To convert from scientific notation to decimal, start by moving the decimal point in the scientific notation number the same number of places as the exponent on the 10. For example, to convert 5.00 * 10 ^− 12 to decimal, you would move the decimal point 12 places to the right, so the number becomes 0.000000000005.
Step-by-step explanation: