Evaluate The Expression A Step By Step Solution

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Introduction

This article provides a step-by-step guide to evaluating the mathematical expression:

2×105(5×10−3)23−(4×103)−(4×102){ \sqrt[3]{\frac{2 \times 10^{5}}{(5 \times 10^{-3})^{2}}} - \sqrt{(4 \times 10^{3}) - (4 \times 10^{2})} }

We will break down the expression, simplify each part, and arrive at the final answer. This guide aims to help you understand the process thoroughly, making it easier to tackle similar problems in the future. The problem involves the application of arithmetic operations, exponents, scientific notation, and roots. Mastering these concepts is crucial for success in mathematics.

Step 1: Simplify the Fraction Inside the Cube Root

The first part of the expression we need to evaluate is the fraction inside the cube root. The expression is:

2×105(5×10−3)2{ \frac{2 \times 10^{5}}{(5 \times 10^{-3})^{2}} }

Let’s start by simplifying the denominator. We have:

(5×10−3)2=52×(10−3)2{ (5 \times 10^{-3})^{2} = 5^{2} \times (10^{-3})^{2} }

Using the rule of exponents, (am)n=amn{(a^{m})^{n} = a^{mn}}, we get:

52×10−6=25×10−6{ 5^{2} \times 10^{-6} = 25 \times 10^{-6} }

Now, substitute this back into the fraction:

2×10525×10−6{ \frac{2 \times 10^{5}}{25 \times 10^{-6}} }

To simplify further, we can divide 2 by 25:

225=0.08{ \frac{2}{25} = 0.08 }

Now, let's handle the exponents. When dividing numbers with the same base, we subtract the exponents:

10510−6=105−(−6)=105+6=1011{ \frac{10^{5}}{10^{-6}} = 10^{5 - (-6)} = 10^{5 + 6} = 10^{11} }

So, the fraction becomes:

0.08×1011{ 0.08 \times 10^{11} }

To express this in scientific notation, we can write 0.08 as 8×10−2{8 \times 10^{-2}}. Therefore, the fraction simplifies to:

8×10−2×1011=8×109{ 8 \times 10^{-2} \times 10^{11} = 8 \times 10^{9} }

This simplified form makes it easier to work with the cube root in the next step. Remember, the goal is to break down the problem into manageable parts, making each step clear and concise. Understanding the rules of exponents and scientific notation is essential for this process.

Step 2: Calculate the Cube Root

Now that we've simplified the fraction inside the cube root, the next step is to calculate the cube root of the simplified expression. From the previous step, we have:

8×1093{ \sqrt[3]{8 \times 10^{9}} }

The cube root of a product can be expressed as the product of the cube roots. Therefore, we can rewrite the expression as:

83×1093{ \sqrt[3]{8} \times \sqrt[3]{10^{9}} }

We know that the cube root of 8 is 2 because 23=8{2^{3} = 8}. So, we have:

83=2{ \sqrt[3]{8} = 2 }

For the cube root of 109{10^{9}}, we can use the property amn=amn{\sqrt[n]{a^{m}} = a^{\frac{m}{n}}} where n is the root. In this case, n = 3 and m = 9. Thus, we get:

1093=1093=103{ \sqrt[3]{10^{9}} = 10^{\frac{9}{3}} = 10^{3} }

Therefore, the cube root part of the expression becomes:

2×103{ 2 \times 10^{3} }

This simplifies the first part of the original expression significantly. The key here is to understand how to apply the properties of roots and exponents to simplify complex expressions. Breaking down the problem into smaller, manageable parts allows for a clearer understanding and reduces the chance of errors.

Step 3: Simplify the Expression Inside the Square Root

Next, we need to simplify the expression inside the square root. The expression is:

(4×103)−(4×102){ \sqrt{(4 \times 10^{3}) - (4 \times 10^{2})} }

First, let's rewrite the terms inside the square root:

4×103=4000{ 4 \times 10^{3} = 4000 }

4×102=400{ 4 \times 10^{2} = 400 }

Now, substitute these values back into the expression inside the square root:

4000−400{ 4000 - 400 }

Perform the subtraction:

4000−400=3600{ 4000 - 400 = 3600 }

So, the expression inside the square root simplifies to 3600. This makes it easier to calculate the square root in the next step. Simplifying expressions by performing arithmetic operations first is a common strategy in mathematics, as it reduces complexity and makes the subsequent steps more straightforward.

Step 4: Calculate the Square Root

Now that we've simplified the expression inside the square root to 3600, we need to calculate the square root of this value:

3600{ \sqrt{3600} }

We can recognize that 3600 is a perfect square. Specifically:

3600=36×100{ 3600 = 36 \times 100 }

Taking the square root of 36 gives us 6, and taking the square root of 100 gives us 10. Therefore, we can write:

3600=36×100=36×100=6×10=60{ \sqrt{3600} = \sqrt{36 \times 100} = \sqrt{36} \times \sqrt{100} = 6 \times 10 = 60 }

So, the square root of 3600 is 60. This step demonstrates the importance of recognizing perfect squares and how they simplify calculations. Understanding basic square roots and their properties can significantly speed up problem-solving in mathematics.

Step 5: Combine the Results

Now that we have simplified both parts of the original expression, we can combine the results. From Step 2, we found that:

2×105(5×10−3)23=2×103=2000{ \sqrt[3]{\frac{2 \times 10^{5}}{(5 \times 10^{-3})^{2}}} = 2 \times 10^{3} = 2000 }

And from Step 4, we found that:

(4×103)−(4×102)=60{ \sqrt{(4 \times 10^{3}) - (4 \times 10^{2})} = 60 }

The original expression is:

2×105(5×10−3)23−(4×103)−(4×102){ \sqrt[3]{\frac{2 \times 10^{5}}{(5 \times 10^{-3})^{2}}} - \sqrt{(4 \times 10^{3}) - (4 \times 10^{2})} }

Substitute the simplified values into the expression:

2000−60{ 2000 - 60 }

Perform the subtraction:

2000−60=1940{ 2000 - 60 = 1940 }

Therefore, the final result of the expression is 1940. This step brings together all the individual simplifications we've made, highlighting the importance of a systematic approach to solving complex problems. By breaking the problem down into smaller parts, we were able to handle each part separately and then combine the results to get the final answer.

Conclusion

In conclusion, by systematically breaking down the expression and applying the rules of exponents, roots, and arithmetic operations, we found that:

2×105(5×10−3)23−(4×103)−(4×102)=1940{ \sqrt[3]{\frac{2 \times 10^{5}}{(5 \times 10^{-3})^{2}}} - \sqrt{(4 \times 10^{3}) - (4 \times 10^{2})} = 1940 }

Therefore, the correct answer is E) 1940. This detailed walkthrough illustrates the importance of methodical problem-solving and a strong understanding of mathematical principles. By practicing these techniques, you can enhance your ability to tackle complex mathematical problems with confidence.

This comprehensive guide should help you understand each step involved in evaluating the given expression. Remember, the key is to break down the problem into manageable parts and apply the appropriate mathematical rules and properties. Understanding these concepts will not only help you solve this specific problem but also equip you with the skills to tackle similar mathematical challenges. Keep practicing and building your mathematical foundation.