Simplifying Expressions with Decimal Exponents

Oct 13, 2025 by magicalceylontours.com 112 views

Understanding Exponents and Expressions

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. The expression given in the problem, $3^{7.869}$ , involves an exponent with a decimal value. To evaluate this expression, we need to understand the properties of exponents and how to simplify expressions with decimal exponents.

Properties of Exponents

Exponents have several properties that help us simplify expressions. One of the most important properties is the product of powers property, which states that when we multiply two numbers with the same base, we can add their exponents. For example, $a^m \cdot a^n = a^{m+n}$ .

Another important property is the power of a power property, which states that when we raise a power to another power, we can multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .

Simplifying Expressions with Decimal Exponents

When we have an expression with a decimal exponent, we can simplify it by converting the decimal to a fraction. For example, $3^{7.869}$ can be simplified as $3^{7+\frac{8}{10}+\frac{6}{100}+\frac{9}{1000}}$ .

To simplify this expression, we need to convert the decimal to a fraction. We can do this by writing the decimal as a sum of fractions with powers of 10 as the denominators. In this case, we can write $7.869$ as $7+\frac{8}{10}+\frac{6}{100}+\frac{9}{1000}$ .

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it by applying the properties of exponents. We can start by evaluating the expression inside the parentheses, which is $7+\frac{8}{10}+\frac{6}{100}+\frac{9}{1000}$ .

To evaluate this expression, we can start by adding the fractions with the same denominator. In this case, we can add the fractions with the denominator 10, which gives us $\frac{80}{10}+\frac{6}{100}+\frac{9}{1000}$ .

Next, we can add the fractions with the denominator 100, which gives us $\frac{80}{10}+\frac{60}{100}+\frac{9}{1000}$ .

Finally, we can add the fractions with the denominator 1000, which gives us $\frac{80}{10}+\frac{60}{100}+\frac{90}{1000}$ .

Simplifying the Expression

Now that we have evaluated the expression inside the parentheses, we can simplify the expression by applying the properties of exponents. We can start by applying the product of powers property, which states that when we multiply two numbers with the same base, we can add their exponents.

In this case, we can multiply the numbers with the same base, which gives us $3^{7+\frac{8}{10}+\frac{6}{100}+\frac{9}{1000}} = 3^7 \cdot 3^{\frac{8}{10}} \cdot 3^{\frac{6}{100}} \cdot 3^{\frac{9}{1000}}$ .

Conclusion

In conclusion, the expression $3^{7.869}$ is equivalent to the expression $3^7 \cdot 3^{\frac{8}{10}} \cdot 3^{\frac{6}{100}} \cdot 3^{\frac{9}{1000}}$ . This is because we can simplify the expression by converting the decimal to a fraction and applying the properties of exponents.

Answer

The correct answer is C. $3^7 \cdot 3^{\frac{86}{10}}$ is not the correct answer because it is not equivalent to the expression $3^{7.869}$ . The correct answer is $3^7 \cdot 3^{\frac{8}{10}} \cdot 3^{\frac{6}{100}} \cdot 3^{\frac{9}{1000}}$ .

Discussion

This problem requires the student to understand the properties of exponents and how to simplify expressions with decimal exponents. The student must be able to convert the decimal to a fraction and apply the properties of exponents to simplify the expression.

The student must also be able to evaluate the expression inside the parentheses and simplify the expression by applying the properties of exponents. This requires a strong understanding of the properties of exponents and how to apply them to simplify expressions.

Additional Examples

Here are some additional examples of expressions with decimal exponents:

$2^{4.567}$
$5^{3.219}$
$7^{2.345}$

These expressions can be simplified by converting the decimal to a fraction and applying the properties of exponents. The student must be able to evaluate the expression inside the parentheses and simplify the expression by applying the properties of exponents.

Practice Problems

Here are some practice problems for the student to try:

Simplify the expression $4^{3.567}$
Simplify the expression $6^{2.345}$
Simplify the expression $8^{4.219}$

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is written above and to the right of a number, indicating how many times the base number should be multiplied by itself. A power, on the other hand, is the result of raising a base number to a certain exponent.

Q: How do I simplify an expression with a decimal exponent?

A: To simplify an expression with a decimal exponent, you need to convert the decimal to a fraction. You can do this by writing the decimal as a sum of fractions with powers of 10 as the denominators.

Q: What is the product of powers property?

A: The product of powers property states that when you multiply two numbers with the same base, you can add their exponents. For example, $a^m \cdot a^n = a^{m+n}$ .

Q: What is the power of a power property?

A: The power of a power property states that when you raise a power to another power, you can multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$ .

Q: How do I evaluate an expression with multiple exponents?

A: To evaluate an expression with multiple exponents, you need to apply the product of powers property and the power of a power property. You can start by evaluating the expression inside the parentheses and then simplify the expression by applying the properties of exponents.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times. A negative exponent indicates that the base number should be divided by itself a certain number of times.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to rewrite the expression with a positive exponent. You can do this by taking the reciprocal of the base number and changing the sign of the exponent.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any number raised to the power of zero is equal to 1. For example, $a^0 = 1$ .

Q: How do I simplify an expression with a zero exponent?

A: To simplify an expression with a zero exponent, you can simply rewrite the expression as 1.

Q: What is the one exponent rule?

A: The one exponent rule states that any number raised to the power of 1 is equal to the number itself. For example, $a^1 = a$ .

Q: How do I simplify an expression with a one exponent?

A: To simplify an expression with a one exponent, you can simply rewrite the expression as the base number itself.

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

A: An exponential expression is an expression that involves raising a base number to a certain exponent. A polynomial expression, on the other hand, is an expression that involves adding or subtracting terms with different exponents.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you need to apply the properties of exponents and simplify the expression by combining like terms.

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

A: An exponential function is a function that involves raising a base number to a certain exponent. A polynomial function, on the other hand, is a function that involves adding or subtracting terms with different exponents.

Q: How do I graph an exponential function?

A: To graph an exponential function, you need to identify the base number and the exponent. You can then use a graphing calculator or a graphing software to graph the function.

Q: What is the difference between an exponential growth and an exponential decay?

A: Exponential growth occurs when a quantity increases at a rate that is proportional to the current value of the quantity. Exponential decay occurs when a quantity decreases at a rate that is proportional to the current value of the quantity.

Q: How do I model real-world situations using exponential functions?

A: To model real-world situations using exponential functions, you need to identify the base number and the exponent. You can then use the exponential function to model the situation and make predictions about the future behavior of the quantity.

Q: What are some common applications of exponential functions?

A: Exponential functions have many common applications in real-world situations, including population growth, chemical reactions, and financial modeling.

Q: How do I use exponential functions to solve problems in finance?

A: To use exponential functions to solve problems in finance, you need to identify the base number and the exponent. You can then use the exponential function to model the situation and make predictions about the future behavior of the quantity.

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

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

Not simplifying the expression by combining like terms
Not applying the properties of exponents correctly
Not identifying the base number and the exponent correctly
Not using the correct notation for exponential functions

Q: How do I choose the correct notation for exponential functions?

A: To choose the correct notation for exponential functions, you need to identify the base number and the exponent. You can then use the correct notation to represent the exponential function.

Q: What are some common applications of exponential functions in science?

A: Exponential functions have many common applications in science, including population growth, chemical reactions, and radioactive decay.

Q: How do I use exponential functions to model real-world situations in science?

A: To use exponential functions to model real-world situations in science, you need to identify the base number and the exponent. You can then use the exponential function to model the situation and make predictions about the future behavior of the quantity.

Q: What are some common mistakes to avoid when working with exponential functions in science?

A: Some common mistakes to avoid when working with exponential functions in science include:

Not simplifying the expression by combining like terms
Not applying the properties of exponents correctly
Not identifying the base number and the exponent correctly
Not using the correct notation for exponential functions

Q: How do I choose the correct notation for exponential functions in science?

A: To choose the correct notation for exponential functions in science, you need to identify the base number and the exponent. You can then use the correct notation to represent the exponential function.

Q: What are some common applications of exponential functions in engineering?

A: Exponential functions have many common applications in engineering, including population growth, chemical reactions, and financial modeling.

Q: How do I use exponential functions to model real-world situations in engineering?

A: To use exponential functions to model real-world situations in engineering, you need to identify the base number and the exponent. You can then use the exponential function to model the situation and make predictions about the future behavior of the quantity.

Q: What are some common mistakes to avoid when working with exponential functions in engineering?

A: Some common mistakes to avoid when working with exponential functions in engineering include:

Not simplifying the expression by combining like terms
Not applying the properties of exponents correctly
Not identifying the base number and the exponent correctly
Not using the correct notation for exponential functions

Q: How do I choose the correct notation for exponential functions in engineering?

A: To choose the correct notation for exponential functions in engineering, you need to identify the base number and the exponent. You can then use the correct notation to represent the exponential function.