Common Mistakes in Simplifying Expressions

Oct 13, 2025 by windailys.com 19 views

Understanding the Problem

In mathematics, simplifying expressions is a crucial step in solving problems. However, mistakes can occur, and it's essential to identify them to ensure accuracy. In this article, we will analyze a given expression and determine which step includes the mistake.

The Expression

The given expression is:

$\frac{1+3^2}{5}+|-10| \div 2$

Step 1: Evaluating the Exponents

The first step in simplifying the expression is to evaluate the exponents. In this case, we have $3^2$ , which equals $9$ . Therefore, the expression becomes:

$\frac{1+9}{5}+|-10| \div 2$

Step 2: Evaluating the Absolute Value

The next step is to evaluate the absolute value of $-10$ , which is $10$ . Therefore, the expression becomes:

$\frac{1+9}{5}+10 \div 2$

Step 3: Evaluating the Division

Now, we need to evaluate the division operation. In this case, we have $10 \div 2$ , which equals $5$ . Therefore, the expression becomes:

$\frac{1+9}{5}+5$

Step 4: Evaluating the Addition

Finally, we need to evaluate the addition operation. In this case, we have $\frac{1+9}{5}+5$ , which equals $\frac{10}{5}+5$ . Simplifying further, we get:

$2+5$

The Correct Answer

The correct answer is:

$7$

Identifying the Mistake

Now that we have the correct answer, let's go back to the original expression and identify the mistake. The original expression is:

$\frac{1+3^2}{5}+|-10| \div 2$

The mistake occurred in Step 2, where we evaluated the absolute value of $-10$ as $10$ . However, the correct evaluation of the absolute value is $|-10| = 10$ , but the mistake was in the order of operations. The correct order of operations is to evaluate the exponents first, then the absolute value, and finally the division.

Correcting the Mistake

To correct the mistake, we need to re-evaluate the expression using the correct order of operations. The correct evaluation of the expression is:

$\frac{1+3^2}{5}+|-10| \div 2$

$= \frac{1+9}{5}+|-10| \div 2$

$= \frac{10}{5}+|-10| \div 2$

$= 2+|-10| \div 2$

$= 2+10 \div 2$

$= 2+5$

$= 7$

Conclusion

In conclusion, the mistake in simplifying the expression occurred in Step 2, where we evaluated the absolute value of $-10$ as $10$ . However, the correct evaluation of the absolute value is $|-10| = 10$ , but the mistake was in the order of operations. By re-evaluating the expression using the correct order of operations, we arrived at the correct answer, which is $7$ .

Common Mistakes in Simplifying Expressions

When simplifying expressions, it's essential to follow the correct order of operations. Some common mistakes include:

Evaluating the absolute value before the exponents
Evaluating the division before the addition
Not following the correct order of operations

Tips for Simplifying Expressions

To simplify expressions accurately, follow these tips:

Evaluate the exponents first
Evaluate the absolute value next
Evaluate the division and addition operations last
Follow the correct order of operations

By following these tips and being aware of common mistakes, you can simplify expressions accurately and arrive at the correct answer.

Practice Problems

To practice simplifying expressions, try the following problems:

Simplify the expression: $\frac{2+4^2}{3}+|-12| \div 2$
Simplify the expression: $\frac{3+2^2}{4}+|-8| \div 3$
Simplify the expression: $\frac{1+5^2}{2}+|-15| \div 4$

Answer Key

$7$
$5$
$10$

Conclusion

Q: What is the correct order of operations when simplifying expressions?

A: The correct order of operations when simplifying expressions is:

Evaluate the exponents (e.g., $2^3$ , $3^2$ )
Evaluate the absolute value (e.g., $|-10|$ , $|5|$ )
Evaluate the multiplication and division operations (e.g., $2 \times 3$ , $10 \div 2$ )
Evaluate the addition and subtraction operations (e.g., $2 + 3$ , $5 - 2$ )

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

A: An exponent is a small number that is raised to a power, while a power is the result of raising a number to a certain exponent. For example, in the expression $2^3$ , the exponent is $3$ and the power is $8$ .

Q: How do I evaluate an absolute value?

A: To evaluate an absolute value, you need to determine the distance of a number from zero on the number line. If the number is positive, the absolute value is the same as the number. If the number is negative, the absolute value is the positive version of the number. For example, $|-10| = 10$ and $|5| = 5$ .

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers, while a decimal is a way of expressing a fraction as a number with a point and digits after the point. For example, the fraction $\frac{1}{2}$ is equal to the decimal $0.5$ .

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD. For example, the fraction $\frac{6}{8}$ can be simplified by dividing both numbers by $2$ to get $\frac{3}{4}$ .

Q: What is the difference between an expression and an equation?

A: An expression is a group of numbers, variables, and mathematical operations that can be evaluated to a value, while an equation is a statement that says two expressions are equal. For example, the expression $2x + 3$ is a group of numbers and variables, while the equation $2x + 3 = 5$ is a statement that says two expressions are equal.

Q: How do I solve an equation?

A: To solve an equation, you need to isolate the variable on one side of the equation by performing inverse operations. For example, to solve the equation $2x + 3 = 5$ , you can subtract $3$ from both sides to get $2x = 2$ , and then divide both sides by $2$ to get $x = 1$ .

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is $1$ , while a quadratic equation is an equation in which the highest power of the variable is $2$ . For example, the equation $2x + 3 = 5$ is a linear equation, while the equation $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. You can then draw a line through the two points to represent the equation. For example, the equation $y = 2x + 3$ can be graphed by finding two points on the line, such as $(0, 3)$ and $(1, 5)$ , and plotting them on a coordinate plane.

Q: What is the difference between a function and a relation?

A: A function is a relation in which each input corresponds to exactly one output, while a relation is a set of ordered pairs that may or may not be a function. For example, the relation $\{(2, 4), (3, 9), (4, 16)\}$ is a function, while the relation $\{(2, 4), (2, 9), (3, 16)\}$ is not a function.

Q: How do I determine if a relation is a function?

A: To determine if a relation is a function, you need to check if each input corresponds to exactly one output. If each input corresponds to exactly one output, then the relation is a function. If not, then the relation is not a function. For example, the relation $\{(2, 4), (3, 9), (4, 16)\}$ is a function because each input corresponds to exactly one output, while the relation $\{(2, 4), (2, 9), (3, 16)\}$ is not a function because the input $2$ corresponds to two different outputs, $4$ and $9$ .