Simplifying the First Square

Oct 13, 2025 by windailys.com 44 views

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying the expression $\left(\frac{2}{5}\right)^2 - \left(\frac{1}{5}\right)^2$ . We will break down the problem step by step, using basic algebraic concepts and techniques to arrive at the correct solution.

Understanding the Problem

The given expression is a difference of two squares, which can be simplified using the formula $a^2 - b^2 = (a + b)(a - b)$ . However, before we can apply this formula, we need to simplify each square separately.

Simplifying the First Square

The first square is $\left(\frac{2}{5}\right)^2$ . To simplify this expression, we need to multiply the numerator and denominator by themselves.

$\left(\frac{2}{5}\right)^2 = \frac{2 \times 2}{5 \times 5} = \frac{4}{25}$

Simplifying the Second Square

The second square is $\left(\frac{1}{5}\right)^2$ . To simplify this expression, we need to multiply the numerator and denominator by themselves.

$\left(\frac{1}{5}\right)^2 = \frac{1 \times 1}{5 \times 5} = \frac{1}{25}$

Subtracting the Two Squares

Now that we have simplified each square, we can subtract the two expressions.

$\frac{4}{25} - \frac{1}{25} = \frac{4 - 1}{25} = \frac{3}{25}$

Choosing the Correct Answer

Based on our calculations, the correct answer is $\frac{3}{25}$ . However, this option is not available in the given choices. Let's re-examine the choices and see if we can find a match.

Re-examining the Choices

Choice A is $\frac{15}{25}$ , which is not equal to our calculated answer. Choice B is $\frac{3}{5}$ , which is also not equal to our calculated answer. Choice C is $\frac{17}{25}$ , which is not equal to our calculated answer. Choice D is $\frac{7}{25}$ , which is not equal to our calculated answer.

Conclusion

Based on our calculations, the correct answer is $\frac{3}{25}$ . However, this option is not available in the given choices. It's possible that there is a mistake in the given choices or that the question is asking for a different type of answer. In any case, the correct answer is $\frac{3}{25}$ .

Final Answer

The final answer is $\boxed{\frac{3}{25}}$ .

Additional Tips and Tricks

When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS).
When subtracting fractions, it's essential to have a common denominator.
When simplifying squares, it's essential to multiply the numerator and denominator by themselves.

Common Mistakes to Avoid

Not following the order of operations (PEMDAS).
Not having a common denominator when subtracting fractions.
Not multiplying the numerator and denominator by themselves when simplifying squares.

Real-World Applications

Simplifying algebraic expressions is an essential skill in many real-world applications, including:

Science: Simplifying algebraic expressions is crucial in scientific calculations, such as calculating the trajectory of a projectile or the motion of an object.
Engineering: Simplifying algebraic expressions is essential in engineering calculations, such as designing bridges or buildings.
Finance: Simplifying algebraic expressions is crucial in financial calculations, such as calculating interest rates or investment returns.

Conclusion

Introduction

In our previous article, we discussed how to simplify the expression $\left(\frac{2}{5}\right)^2 - \left(\frac{1}{5}\right)^2$ . We broke down the problem step by step, using basic algebraic concepts and techniques to arrive at the correct solution. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying an algebraic expression. PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next (e.g., $2^3$ ).
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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 denominator. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder. Once you have found the GCD, you can divide both the numerator and denominator by the GCD to simplify the fraction.

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$ . This formula can be used to simplify expressions of the form $a^2 - b^2$ .

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. Once you have found the largest perfect square, you can simplify the square root by dividing the number inside the square root by the perfect square.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. The GCD can be found using the Euclidean algorithm or by listing the factors of each number and finding the largest common factor.

Q: How do I simplify an expression with multiple terms?

A: To simplify an expression with multiple terms, you need to combine like terms. Like terms are terms that have the same variable raised to the same power. You can combine like terms by adding or subtracting their coefficients.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. A constant is a value that does not change. In algebraic expressions, variables are often represented by letters such as x, y, or z, while constants are represented by numbers.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to follow the rules of exponentiation. The rules of exponentiation are:

$a^m \cdot a^n = a^{m+n}$
$(a^m)^n = a^{m \cdot n}$
$a^m \div a^n = a^{m-n}$

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it has many real-world applications. By following the order of operations (PEMDAS), simplifying fractions, using the difference of squares formula, simplifying square roots, finding the greatest common divisor (GCD), combining like terms, and following the rules of exponentiation, we can simplify even the most complex algebraic expressions.

Additional Tips and Tricks

Always follow the order of operations (PEMDAS) when simplifying algebraic expressions.
Simplify fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
Use the difference of squares formula to simplify expressions of the form $a^2 - b^2$ .
Simplify square roots by finding the largest perfect square that divides the number inside the square root.
Combine like terms by adding or subtracting their coefficients.
Follow the rules of exponentiation to simplify expressions with exponents.

Common Mistakes to Avoid

Not following the order of operations (PEMDAS) when simplifying algebraic expressions.
Not simplifying fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
Not using the difference of squares formula to simplify expressions of the form $a^2 - b^2$ .
Not simplifying square roots by finding the largest perfect square that divides the number inside the square root.
Not combining like terms by adding or subtracting their coefficients.
Not following the rules of exponentiation to simplify expressions with exponents.