Evaluate $\frac{1.4 \times 10^{-4} \times 2.1 \times 10^3}{2.8 \times 10^4}$ leaving your answer in standard form.
Answer (2)
To evaluate the expression 2.8 × 1 0 4 1.4 × 1 0 − 4 × 2.1 × 1 0 3 , separate the numerical part and the powers of ten. This leads us to 1.05 × 1 0 − 5 , which is already in standard form.
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Rewrite the expression as a product of numerical part and powers of 10.
Simplify the numerical part: 2.8 1.4 × 2.1 = 1.05 .
Simplify the powers of 10: 1 0 4 1 0 − 4 × 1 0 3 = 1 0 − 5 .
Combine the results to get the final answer in standard form: 1.05 × 1 0 − 5 .
Explanation
Understanding the problem We are asked to evaluate the expression 2.8 × 1 0 4 1.4 × 1 0 − 4 × 2.1 × 1 0 3 and express the answer in standard form. Standard form means we want to write the answer as a × 1 0 b , where 1 ≤ a < 10 and b is an integer.
Rewriting the expression First, let's rewrite the expression to separate the numbers from the powers of 10: 2.8 × 1 0 4 1.4 × 1 0 − 4 × 2.1 × 1 0 3 = 2.8 1.4 × 2.1 × 1 0 4 1 0 − 4 × 1 0 3
Simplifying the numerical part Now, let's simplify the numerical part: 2.8 1.4 × 2.1 = 2.8 2.94 = 1.05
Simplifying the powers of 10 Next, we simplify the powers of 10 using exponent rules. Recall that 1 0 m × 1 0 n = 1 0 m + n and 1 0 n 1 0 m = 1 0 m − n . Thus, 1 0 4 1 0 − 4 × 1 0 3 = 1 0 4 1 0 − 4 + 3 = 1 0 4 1 0 − 1 = 1 0 − 1 − 4 = 1 0 − 5
Combining the results Now, we combine the simplified numerical part and the simplified power of 10: 1.05 × 1 0 − 5 Since 1 ≤ 1.05 < 10 , this is already in standard form.
Final Answer Therefore, the final answer in standard form is 1.05 × 1 0 − 5 .
Examples
Standard form is useful in many scientific applications. For example, the size of a bacteria is very small and is better represented in standard form. Similarly, astronomical distances are very large and are better represented using standard form. Using standard form helps in comparing very large and very small quantities easily.
