Select The Best Answer For The Question.1. What Is The Value Of 12.5 × 10 7 12.5 \times 10^7 12.5 × 1 0 7 ?A. 1,250,000 B. 125,000,000 C. 1.2500000 D. 12.5000000

Mar 9, 2025 by ADMIN 165 views

**Understanding the Value of Exponential Expressions**

When dealing with exponential expressions, it's essential to understand the concept of scientific notation and how to evaluate expressions with exponents. In this article, we'll explore the value of the expression $12.5 \times 10^7$ and determine the correct answer among the given options.

What is Scientific Notation?

Scientific notation is a way of expressing numbers in the form $a \times 10^b$ , where $a$ is a number between 1 and 10, and $b$ is an integer. This notation is commonly used to represent very large or very small numbers in a more compact and manageable form.

Evaluating Exponential Expressions

To evaluate an exponential expression, we need to multiply the coefficient by the base raised to the power of the exponent. In the case of $12.5 \times 10^7$ , we have a coefficient of 12.5 and an exponent of 7.

Step 1: Multiply the Coefficient by the Base

To evaluate the expression, we need to multiply 12.5 by $10^7$ . Since $10^7$ is equal to 10,000,000, we can multiply 12.5 by this value.

result = 12.5 * 10**7

Step 2: Calculate the Result

When we multiply 12.5 by 10,000,000, we get a result of 125,000,000.

result = 12.5 * 10,000,000
result = 125,000,000

Conclusion

Based on the evaluation of the expression $12.5 \times 10^7$ , we can conclude that the correct answer is:

B. 125,000,000

This answer is the result of multiplying the coefficient 12.5 by the base $10^7$ , which is equal to 10,000,000.

Why is this Answer Correct?

The answer B. 125,000,000 is correct because it is the result of evaluating the expression $12.5 \times 10^7$ using the rules of scientific notation and exponentiation. The other options, A. 1,250,000, C. 1.2500000, and D. 12.5000000, are incorrect because they do not accurately represent the result of evaluating the expression.

Common Mistakes to Avoid

When evaluating exponential expressions, it's essential to avoid common mistakes such as:

Forgetting to multiply the coefficient by the base raised to the power of the exponent
Not using the correct rules of exponentiation
Not converting the expression to scientific notation

By avoiding these mistakes, you can ensure that you get the correct answer when evaluating exponential expressions.

Real-World Applications

Exponential expressions have many real-world applications, such as:

Calculating the area and volume of shapes
Modeling population growth and decay
Representing very large or very small numbers in a compact form

In this article, we'll answer some frequently asked questions about exponential expressions, including their definition, rules, and applications.

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that involves a base raised to a power, such as $2^3$ or $10^5$ . Exponential expressions are used to represent very large or very small numbers in a compact form.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression involves a base raised to a power, while a polynomial expression involves a sum of terms with different powers of a variable. For example, $2^3$ is an exponential expression, while $2x^2 + 3x + 1$ is a polynomial expression.

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, you need to multiply the base by itself as many times as the exponent indicates. For example, to evaluate $2^3$ , you would multiply 2 by itself three times: $2 \times 2 \times 2 = 8$ .

Q: What is the order of operations for exponential expressions?

A: The order of operations for exponential expressions is the same as for other mathematical expressions: parentheses, exponents, multiplication and division, and addition and subtraction. For example, to evaluate the expression $2^3 + 4 \times 5$ , you would first evaluate the exponentiation, then the multiplication, and finally the addition.

Q: Can I simplify an exponential expression?

A: Yes, you can simplify an exponential expression by combining like terms or using the rules of exponentiation. For example, the expression $2^3 \times 2^2$ can be simplified to $2^5$ using the rule of exponentiation that states $a^m \times a^n = a^{m+n}$ .

Q: What are some common mistakes to avoid when working with exponential expressions?

A: Some common mistakes to avoid when working with exponential expressions include:

Forgetting to multiply the base by itself as many times as the exponent indicates
Not using the correct order of operations
Not simplifying the expression using the rules of exponentiation
Not converting the expression to scientific notation

Q: How do I convert an exponential expression to scientific notation?

A: To convert an exponential expression to scientific notation, you need to express the base as a number between 1 and 10, and the exponent as a power of 10. For example, the expression $2^3$ can be converted to scientific notation as $2.0 \times 10^3$ .

Q: What are some real-world applications of exponential expressions?

A: Exponential expressions have many real-world applications, including:

Calculating the area and volume of shapes
Modeling population growth and decay
Representing very large or very small numbers in a compact form
Solving problems in finance, economics, and science

Q: Can I use a calculator to evaluate exponential expressions?

A: Yes, you can use a calculator to evaluate exponential expressions. However, it's essential to understand the rules of exponentiation and how to evaluate expressions manually to ensure accuracy and to develop problem-solving skills.

Q: How do I determine the value of an exponential expression with a negative exponent?

A: To determine the value of an exponential expression with a negative exponent, you need to use the rule of exponentiation that states $a^{-m} = \frac{1}{a^m}$ . For example, the expression $2^{-3}$ can be evaluated as $\frac{1}{2^3} = \frac{1}{8}$ .

Q: Can I use exponential expressions to solve problems in finance and economics?

A: Yes, you can use exponential expressions to solve problems in finance and economics, such as calculating compound interest, modeling population growth, and analyzing economic data.

Q: How do I use exponential expressions to model population growth?

A: To use exponential expressions to model population growth, you need to use the rule of exponentiation that states $a^m \times a^n = a^{m+n}$ . For example, if the population of a city is growing at a rate of 2% per year, the population after 5 years can be modeled using the expression $P(1 + 0.02)^5$ , where $P$ is the initial population.

Q: Can I use exponential expressions to solve problems in science?

A: Yes, you can use exponential expressions to solve problems in science, such as calculating the half-life of a radioactive substance, modeling the growth of a population, and analyzing data from experiments.

Q: How do I use exponential expressions to calculate the half-life of a radioactive substance?

A: To use exponential expressions to calculate the half-life of a radioactive substance, you need to use the rule of exponentiation that states $a^m \times a^n = a^{m+n}$ . For example, if the half-life of a radioactive substance is 5 years, the amount of the substance remaining after 10 years can be modeled using the expression $A(1/2)^2$ , where $A$ is the initial amount of the substance.