Identifying Terms with Integer Exponents

Oct 13, 2025 by leolavanderia.com.br 21 views

Introduction

In algebra, the exponent of a term in an expression is a crucial concept that helps us understand the behavior of the expression. The exponent of a term is the power to which the variable or constant is raised. In this article, we will focus on determining the integer exponent of the expression $$3x^2 + 5x^{\frac{1}{2}}$$. We will explore the concept of exponents, identify the terms with integer exponents, and determine the correct answer from the given options.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number or variable. For example, $$x^2$$ represents $$x \times x$$, and $$x^3$$ represents $$x \times x \times x$$. The exponent tells us how many times the base (in this case, $$x$$) is multiplied by itself.

Identifying Terms with Integer Exponents

In the given expression $$3x^2 + 5x^\frac{1}{2}}$$, we can see that there are two terms $$3x^2$$ and $$5x^{\frac{1{2}}$$. The first term, $$3x^2$$, has an integer exponent of 2, since the variable $$x$$ is raised to the power of 2. The second term, $$5x^{\frac{1}{2}}$$, has a non-integer exponent of $$\frac{1}{2}$$, since the variable $$x$$ is raised to the power of $$\frac{1}{2}$$.

Determining the Integer Exponent

Since we are asked to determine the integer exponent of the expression, we need to focus on the term with the integer exponent. In this case, the term with the integer exponent is $$3x^2$$, which has an exponent of 2.

Conclusion

In conclusion, the integer exponent of the expression $$3x^2 + 5x^{\frac{1}{2}}$$ is 2. This is because the term $$3x^2$$ has an integer exponent of 2, while the term $$5x^{\frac{1}{2}}$$ has a non-integer exponent of $$\frac{1}{2}$$.

Answer

The correct answer is:

A. 2

Additional Information

The term $$3x^2$$ has an integer exponent of 2, since the variable $$x$$ is raised to the power of 2.
The term $$5x^{\frac{1}{2}}$$ has a non-integer exponent of $$\frac{1}{2}$$, since the variable $$x$$ is raised to the power of $$\frac{1}{2}$$.
The expression $$3x^2 + 5x^\frac{1}{2}}$$ has two terms $$3x^2$$ and $$5x^{\frac{1{2}}$$.
The integer exponent of an algebraic expression is the exponent of the term with the highest power of the variable.

References

Frequently Asked Questions

Q: What is the integer exponent of the expression $$3x^2 + 5x^{\frac{1}{2}}$$?
A: The integer exponent of the expression $$3x^2 + 5x^{\frac{1}{2}}$$ is 2.
Q: Why is the term $$3x^2$$ considered to have an integer exponent?
A: The term $$3x^2$$ has an integer exponent of 2, since the variable $$x$$ is raised to the power of 2.
Q: Why is the term $$5x^{\frac{1}{2}}$$ considered to have a non-integer exponent?
A: The term $$5x^{\frac{1}{2}}$$ has a non-integer exponent of $$\frac{1}{2}$$, since the variable $$x$$ is raised to the power of $$\frac{1}{2}$$.
Determining the Integer Exponent of an Algebraic Expression ===========================================================

Q&A: Determining the Integer Exponent

Q: What is the integer exponent of the expression $$3x^2 + 5x^{\frac{1}{2}}$$?

A: The integer exponent of the expression $$3x^2 + 5x^{\frac{1}{2}}$$ is 2. This is because the term $$3x^2$$ has an integer exponent of 2, while the term $$5x^{\frac{1}{2}}$$ has a non-integer exponent of $$\frac{1}{2}$$.

Q: Why is the term $$3x^2$$ considered to have an integer exponent?

A: The term $$3x^2$$ has an integer exponent of 2, since the variable $$x$$ is raised to the power of 2. In other words, the exponent 2 is an integer, and it is the power to which the variable $$x$$ is raised.

Q: Why is the term $$5x^{\frac{1}{2}}$$ considered to have a non-integer exponent?

A: The term $$5x^{\frac{1}{2}}$$ has a non-integer exponent of $$\frac{1}{2}$$, since the variable $$x$$ is raised to the power of $$\frac{1}{2}$$. In other words, the exponent $$\frac{1}{2}$$ is not an integer, and it is not the power to which the variable $$x$$ is raised.

Q: How do I determine the integer exponent of an algebraic expression?

A: To determine the integer exponent of an algebraic expression, you need to identify the terms with integer exponents. In other words, you need to find the terms where the variable is raised to the power of an integer. Once you have identified the terms with integer exponents, you can determine the integer exponent of the expression.

Q: What is the difference between an integer exponent and a non-integer exponent?

A: An integer exponent is an exponent that is a whole number, such as 2, 3, or 4. A non-integer exponent is an exponent that is not a whole number, such as $$\frac{1}{2}$$, $$\frac{3}{4}$$, or $$\frac{5}{6}$$.

Q: Can an algebraic expression have multiple integer exponents?

A: Yes, an algebraic expression can have multiple integer exponents. For example, the expression $$x^2 + 3x^3 + 5x^4$$ has three integer exponents: 2, 3, and 4.

Q: How do I simplify an algebraic expression with multiple integer exponents?

A: To simplify an algebraic expression with multiple integer exponents, you need to combine like terms. In other words, you need to add or subtract terms that have the same variable and exponent. Once you have combined like terms, you can simplify the expression.

Q: What is the importance of determining the integer exponent of an algebraic expression?

A: Determining the integer exponent of an algebraic expression is important because it helps you understand the behavior of the expression. In other words, it helps you understand how the expression changes when the variable is changed. This is important in many areas of mathematics and science, such as calculus, algebra, and physics.

Q: Can you provide examples of algebraic expressions with integer exponents?

A: Yes, here are some examples of algebraic expressions with integer exponents:

$x^2 + 3x^3 + 5x^4$
$2x^3 - 3x^2 + 4x$
$x^4 + 2x^3 - 3x^2 + 4x$

Q: Can you provide examples of algebraic expressions with non-integer exponents?

A: Yes, here are some examples of algebraic expressions with non-integer exponents:

$x^{\frac{1}{2}} + 3x^{\frac{3}{4}} + 5x^{\frac{5}{6}}$
$2x^{\frac{3}{2}} - 3x^{\frac{1}{2}} + 4x$
$x^{\frac{4}{3}} + 2x^{\frac{3}{4}} - 3x^{\frac{2}{3}} + 4x$

Q: How do I determine the non-integer exponent of an algebraic expression?

A: To determine the non-integer exponent of an algebraic expression, you need to identify the terms with non-integer exponents. In other words, you need to find the terms where the variable is raised to the power of a non-integer. Once you have identified the terms with non-integer exponents, you can determine the non-integer exponent of the expression.

Q: What is the difference between an algebraic expression with integer exponents and an algebraic expression with non-integer exponents?

A: An algebraic expression with integer exponents is an expression where the variable is raised to the power of an integer, such as 2, 3, or 4. An algebraic expression with non-integer exponents is an expression where the variable is raised to the power of a non-integer, such as $$\frac{1}{2}$$, $$\frac{3}{4}$$, or $$\frac{5}{6}$$.

Q: Can an algebraic expression have both integer and non-integer exponents?

A: Yes, an algebraic expression can have both integer and non-integer exponents. For example, the expression $$x^2 + 3x^{\frac{1}{2}} + 5x^4$$ has both integer and non-integer exponents.

Q: How do I simplify an algebraic expression with both integer and non-integer exponents?

A: To simplify an algebraic expression with both integer and non-integer exponents, you need to combine like terms. In other words, you need to add or subtract terms that have the same variable and exponent. Once you have combined like terms, you can simplify the expression.