Problem 103 - Elementary Algebra

Question

\(P=3.9 \times 10^{3} \frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}}-\left(-2.25 \times 10^{4} \frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}}\right)\)

Step-by-Step Solution

Answer

2.64 × 10^4 kg·m/s

Step 1: Understand the Problem

The problem involves subtracting one term from another. Notice that you are given momentum values in scientific notation and one of the terms has a negative sign.

Step 2: Rewrite the Expression

Rewrite the expression to make it clearer: \[ P = 3.9 \times 10^3 \frac{kg \times m}{s} - (-2.25 \times 10^4 \frac{kg \times m}{s}) \]

Step 3: Convert the Double Negative

Realize that subtracting a negative is the same as adding the positive equivalent: \[ P = 3.9 \times 10^3 \frac{kg \times m}{s} + 2.25 \times 10^4 \frac{kg \times m}{s} \]

Step 4: Add the Terms

Since the terms have the same units, add the coefficients: \[ P = (3.9 \times 10^3 + 2.25 \times 10^4) \frac{kg \times m}{s} \]

Step 5: Convert Notations if Needed

Convert one of the terms to match the power of ten of the other term if necessary. Convert one to make both terms similar: \[ 3.9 \times 10^3 = 0.39 \times 10^4 \]

Step 6: Add the Numbers

Now, add the numbers: \[ P = (0.39 \times 10^4 + 2.25 \times 10^4) \frac{kg \times m}{s} = 2.64 \times 10^4 \frac{kg \times m}{s} \]

Step 7: Write the Final Result

Express the final momentum value clearly: \[ P = 2.64 \times 10^4 \frac{kg \times m}{s} \]

Key Concepts

Adding Coefficients

When adding numbers in scientific notation, it's important that the exponents (powers of ten) match. If they don't, convert one of the numbers so they have the same exponent.

In the problem, we have: \[ P = 3.9 \times 10^3 + 2.25 \times 10^4 \] To add these, convert \( 3.9 \times 10^3 \) into \( 0.39 \times 10^4 \). Now, both numbers have the same power of ten: \[ P = 0.39 \times 10^4 + 2.25 \times 10^4 \] Now, simply add the coefficients: \[ P = (0.39 + 2.25) \times 10^4 = 2.64 \times 10^4 \]

Understanding Powers of Ten

Powers of ten are used in scientific notation to simplify the writing and calculation of very large or very small numbers.

The power of ten indicates how many times to multiply the base number (usually 10) by itself. For example, \( 10^3 \) means 10 multiplied by itself three times (10 x 10 x 10 = 1000).

In the problem, we used powers of ten to express momentum values. This not only simplifies our calculations but also makes it easier to understand and compare large numbers. To add or subtract numbers in scientific notation, make sure the powers of ten are the same, then combine the coefficients.