Simplify The Expression: $ (5)(-4)^2 - 15 - 3 $

Mar 13, 2025 by ADMIN 48 views

**Simplify the Expression: $(5)(-4)^2 - 15 - 3$**

Introduction

In this article, we will simplify the given mathematical expression: $(5)(-4)^2 - 15 - 3$ . This expression involves exponentiation, multiplication, and subtraction. We will break down the expression step by step to simplify it.

Understanding the Expression

The given expression is $(5)(-4)^2 - 15 - 3$ . Let's analyze each part of the expression:

$(5)$ : This is a constant term.
$(-4)^2$ : This is an exponentiation operation where $-4$ is raised to the power of $2$ .
$-15$ : This is another constant term.
$-3$ : This is the last constant term.

Step 1: Simplify the Exponentiation

The first step is to simplify the exponentiation operation: $(-4)^2$ . According to the exponentiation rules, when a negative number is raised to an even power, the result is positive.

(-4)^2 = 16

So, the expression becomes: $(5)(16) - 15 - 3$ .

Step 2: Multiply the Constants

Next, we multiply the constant term $(5)$ with the result of the exponentiation operation $(16)$ .

(5)(16) = 80

Now, the expression becomes: $80 - 15 - 3$ .

Step 3: Subtract the Constants

Finally, we subtract the constant terms: $-15$ and $-3$ from the result of the multiplication operation $(80)$ .

80 - 15 = 65
65 - 3 = 62

Therefore, the simplified expression is: $62$ .

Conclusion

In this article, we simplified the given mathematical expression: $(5)(-4)^2 - 15 - 3$ . We broke down the expression step by step, starting with the exponentiation operation, then multiplying the constants, and finally subtracting the constants. The final simplified expression is $62$ .

Key Takeaways

Exponentiation rules state that when a negative number is raised to an even power, the result is positive.
Multiplication of constants can be performed before subtraction.
Subtraction of constants can be performed in any order.

Real-World Applications

Simplifying mathematical expressions is an essential skill in various fields, including:

Science: Simplifying expressions is crucial in scientific calculations, such as calculating the trajectory of a projectile or the energy of a system.
Engineering: Engineers use mathematical expressions to design and optimize systems, such as bridges or electronic circuits.
Finance: Financial analysts use mathematical expressions to calculate investment returns, interest rates, and other financial metrics.

Common Mistakes

When simplifying mathematical expressions, it's essential to avoid common mistakes, such as:

Forgetting to apply exponentiation rules: Failing to apply exponentiation rules can lead to incorrect results.
Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect results.
Not simplifying expressions: Failing to simplify expressions can lead to complex and difficult-to-understand results.

Final Thoughts

Introduction

In our previous article, we simplified the given mathematical expression: $(5)(-4)^2 - 15 - 3$ . We broke down the expression step by step, starting with the exponentiation operation, then multiplying the constants, and finally subtracting the constants. In this article, we will answer some frequently asked questions related to simplifying mathematical expressions.

Q&A

Q: What is the order of operations in mathematics?

A: The order of operations in mathematics is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS is commonly used to remember the order of operations:

P: Parentheses - Evaluate expressions inside parentheses first.
E: Exponents - Evaluate any exponential expressions next.
M: Multiplication - Evaluate any multiplication operations after that.
D: Division - Evaluate any division operations after multiplication.
A: Addition - Evaluate any addition operations after division.
S: Subtraction - Finally, evaluate any subtraction operations.

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

A: To simplify an expression with multiple operations, follow the order of operations (PEMDAS). Start by evaluating any expressions inside parentheses, then evaluate any exponential expressions, followed by multiplication and division operations, and finally addition and subtraction operations.

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

A: In mathematics, a variable is a symbol that represents a value that can change, while a constant is a value that remains the same. For example, in the expression $x + 5$ , $x$ is a variable, while $5$ is a constant.

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

A: To simplify an expression with a negative exponent, follow these steps:

Change the sign of the exponent: Change the sign of the exponent to make it positive.
Take the reciprocal: Take the reciprocal of the base number.
Simplify the expression: Simplify the resulting expression.

For example, to simplify the expression $(-4)^{-2}$ , follow these steps:

Change the sign of the exponent: Change the sign of the exponent to make it positive: $(-4)^2$ .
Take the reciprocal: Take the reciprocal of the base number: $\frac{1}{(-4)^2}$ .
Simplify the expression: Simplify the resulting expression: $\frac{1}{16}$ .

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, follow these steps:

Simplify the numerator and denominator: Simplify the numerator and denominator separately.
Divide the numerator and denominator: Divide the numerator and denominator by their greatest common divisor (GCD).
Simplify the resulting fraction: Simplify the resulting fraction.

For example, to simplify the expression $\frac{12}{18}$ , follow these steps:

Simplify the numerator and denominator: Simplify the numerator and denominator separately: $\frac{6}{9}$ .
Divide the numerator and denominator: Divide the numerator and denominator by their GCD: $\frac{2}{3}$ .
Simplify the resulting fraction: Simplify the resulting fraction: $\frac{2}{3}$ .