Finding The Exponent What Number Should Be The Exponent On The 10
In mathematics, understanding exponents and scientific notation is crucial for expressing very large or very small numbers in a concise and manageable way. This article delves into the concept of exponents, particularly focusing on powers of 10, and how they play a vital role in converting numbers between different forms. We will explore a specific problem that requires determining the correct exponent to equate two given numbers. This involves manipulating decimal points and understanding the relationship between the magnitude of a number and its representation in scientific notation. Exponents are fundamental in various scientific and engineering applications, making this a key concept for students and professionals alike. By the end of this article, you will have a clear understanding of how to solve such problems and the underlying principles that govern them.
Decoding the Equation: 75.6 * 10^y = 0.00756
To solve the equation $75.6 imes 10^y = 0.00756$, our primary goal is to find the value of $y$, which represents the exponent of 10. This exponent will tell us how many places and in which direction (positive or negative) we need to move the decimal point in 75.6 to make it equal to 0.00756. The equation essentially asks: what power of 10 do we need to multiply by 75.6 to get 0.00756? This type of problem is common in scientific notation, where numbers are expressed as a product of a decimal number between 1 and 10 and a power of 10. To solve this, we'll first look at the magnitude change required. We need to reduce 75.6 to 0.00756, which means we are dealing with a negative exponent because we are making the number smaller. We will then count the number of decimal places the decimal point needs to be moved. This will give us the absolute value of $y$. By understanding these steps, we can systematically approach the problem and arrive at the correct solution. The application of scientific notation here is crucial, as it allows us to express very large or small numbers efficiently. The correct value of $y$ will bridge the gap between these two numbers, demonstrating the power and utility of exponents in mathematical problem-solving.
Step-by-Step Solution
Understanding the Movement of the Decimal Point
When dealing with powers of 10, understanding how the decimal point moves is crucial. Multiplying a number by $10^y$ shifts the decimal point $|y|$ places. If $y$ is positive, the decimal point moves to the right, making the number larger. If $y$ is negative, the decimal point moves to the left, making the number smaller. In our problem, we need to transform 75.6 into 0.00756. This transformation clearly involves making the number smaller, indicating that $y$ must be negative. The number of places the decimal point needs to move can be counted directly by observing the two numbers. This direct relationship between the exponent and the decimal point movement is a cornerstone of working with scientific notation. Understanding this concept makes it easier to visualize and solve problems involving powers of 10. Moreover, this skill is essential not only in mathematics but also in various scientific disciplines where large and small numbers are frequently encountered.
Counting Decimal Places
To determine the value of $y$, we need to count how many places the decimal point must move in 75.6 to obtain 0.00756. Start by writing down both numbers: 75.6 and 0.00756. In 75.6, the decimal point is implicitly after the 6. To transform 75.6 into 0.00756, the decimal point needs to move to the left. We count the number of positions: one position moves it to 7.56, two positions to 0.756, three positions to 0.0756, and four positions to 0.00756. Therefore, the decimal point has moved four places to the left. Since the movement is to the left, this indicates a negative exponent. The number of places moved gives us the magnitude of the exponent. Thus, we find that $y$ corresponds to moving the decimal point four places to the left, making it a negative exponent. This careful counting and understanding of direction are crucial for accurately determining the exponent.
Determining the Value of y
From our previous step, we've established that the decimal point needs to move four places to the left to transform 75.6 into 0.00756. This directly translates to a negative exponent of 4. Therefore, $y = -3$ is the correct exponent. This means that $75.6 imes 10^-3}$ should equal 0.00756. We can verify this by performing the multiplication = 75.6 imes 0.001 = 0.0756$. This calculation confirms that our determined value for $y$ is correct. Understanding the sign and magnitude of the exponent is critical in these types of problems. A negative exponent signifies a number less than 1, while a positive exponent indicates a number greater than 1. The exponent's magnitude indicates the scale of the number relative to the base number. In summary, by carefully counting the decimal places and considering the direction of movement, we can accurately find the value of the exponent.
Evaluating the Options
Option A: y = -3
If we substitute $y = -3$ into the equation $75.6 imes 10^y = 0.00756$, we get $75.6 imes 10^{-3}$. This is equivalent to $75.6 imes 0.001$, which equals 0.0756. However, we're trying to achieve 0.00756. Comparing 0.0756 with 0.00756, we can see that they are not equal. Thus, option A is incorrect because it does not satisfy the original equation. It's crucial to meticulously check each option by plugging it back into the original equation to ensure accuracy. Careful evaluation is a key step in problem-solving, especially in mathematics, where a small error can lead to a wrong answer. Understanding the effect of different exponents on the magnitude of a number is also vital in this evaluation process.
Option B: y = 3
Substituting $y = 3$ into the equation $75.6 imes 10^y = 0.00756$ gives us $75.6 imes 10^3$. This is equivalent to $75.6 imes 1000$, which equals 75600. Clearly, 75600 is not equal to 0.00756. Option B is incorrect because it results in a much larger number, whereas we need a much smaller number. This highlights the importance of understanding the direction of decimal movement based on the sign of the exponent. A positive exponent increases the number significantly, moving the decimal point to the right, while a negative exponent decreases the number, moving the decimal point to the left. Therefore, recognizing the effect of a positive exponent in this context immediately rules out this option. Accurate substitution and evaluation are essential for identifying the correct solution.
Option C: y = -4
Let's evaluate option C, where $y = -4$. Substituting this value into the equation $75.6 imes 10^y = 0.00756$, we have $75.6 imes 10^{-4}$. This is the same as $75.6 imes 0.0001$, which equals 0.00756. This result matches the target value of 0.00756. Therefore, option C is the correct answer. This step demonstrates the importance of methodically evaluating each option until the correct one is found. It also underscores the significance of accurate multiplication and decimal placement when working with powers of 10. By substituting and calculating, we can definitively verify whether a particular exponent satisfies the given equation. The process of verification is a fundamental aspect of mathematical problem-solving.
Option D: y = 2
Finally, let's examine option D, where $y = 2$. Substituting this value into the equation $75.6 imes 10^y = 0.00756$, we get $75.6 imes 10^2$. This is equivalent to $75.6 imes 100$, which equals 7560. This result is significantly different from our target value of 0.00756. Option D is therefore incorrect. This evaluation further reinforces the understanding that a positive exponent increases the value of the original number, moving the decimal point to the right. In contrast, we needed to decrease the value substantially to match 0.00756, which requires a negative exponent. This comparative analysis of options highlights the importance of recognizing the impact of the sign and magnitude of the exponent. By systematically evaluating each option, we can confidently eliminate incorrect choices and arrive at the correct solution.
Conclusion: The Correct Exponent
In conclusion, the correct value for the exponent $y$ in the equation $75.6 imes 10^y = 0.00756$ is $y = -4$. This was determined by understanding that the decimal point needed to be moved four places to the left to transform 75.6 into 0.00756. We methodically evaluated each option, substituting the given values of $y$ into the equation and verifying whether the result matched the target value of 0.00756. Only option C, with $y = -4$, satisfied the equation, demonstrating the power of negative exponents in reducing the magnitude of a number. This exercise underscores the importance of understanding exponents, decimal point movement, and the systematic evaluation of options in mathematical problem-solving. Mastery of these concepts is crucial for tackling more complex problems in scientific notation and other areas of mathematics and science.
Exponent, Scientific Notation, Decimal Point, Powers of 10, Mathematical Problem-Solving, Negative Exponent, Evaluation, Equation, Verification
