Simplify The Expression:$\[ \frac{2^2}{2^3} + \frac{7^2}{7^3} = \\]

Feb 28, 2025 by ADMIN 69 views

Simplify the Expression: A Comprehensive Guide to Evaluating Exponential Expressions

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us evaluate complex equations and solve problems efficiently. One of the most common types of expressions that require simplification are exponential expressions, which involve powers of numbers. In this article, we will focus on simplifying the expression $\frac{2^2}{2^3} + \frac{7^2}{7^3}$ , and provide a step-by-step guide on how to evaluate it.

Understanding Exponential Expressions

Exponential expressions are a type of mathematical expression that involves a base number raised to a power. For example, $2^3$ means $2$ raised to the power of $3$ , which is equal to $8$ . Exponential expressions can be simplified using the rules of exponents, which state that when we divide two exponential expressions with the same base, we subtract the exponents.

Simplifying the Expression

To simplify the expression $\frac{2^2}{2^3} + \frac{7^2}{7^3}$ , we need to follow the order of operations (PEMDAS):

Evaluate the exponents: First, we need to evaluate the exponents in the expression. $2^2$ means $2$ raised to the power of $2$ , which is equal to $4$ . Similarly, $2^3$ means $2$ raised to the power of $3$ , which is equal to $8$ . $7^2$ means $7$ raised to the power of $2$ , which is equal to $49$ . $7^3$ means $7$ raised to the power of $3$ , which is equal to $343$ .
Simplify the fractions: Now that we have evaluated the exponents, we can simplify the fractions. $\frac{2^2}{2^3}$ can be simplified as $\frac{4}{8}$ , which is equal to $\frac{1}{2}$ . Similarly, $\frac{7^2}{7^3}$ can be simplified as $\frac{49}{343}$ , which is equal to $\frac{1}{7}$ .
Add the fractions: Now that we have simplified the fractions, we can add them together. $\frac{1}{2} + \frac{1}{7}$ can be added by finding a common denominator, which is $14$ . So, $\frac{1}{2} + \frac{1}{7} = \frac{7}{14} + \frac{2}{14} = \frac{9}{14}$ .

Conclusion

In conclusion, simplifying the expression $\frac{2^2}{2^3} + \frac{7^2}{7^3}$ requires us to follow the order of operations (PEMDAS) and use the rules of exponents. By evaluating the exponents, simplifying the fractions, and adding the fractions, we can arrive at the final answer, which is $\frac{9}{14}$ .

Tips and Tricks

Here are some tips and tricks to help you simplify exponential expressions:

Use the rules of exponents: When dividing two exponential expressions with the same base, subtract the exponents.
Evaluate the exponents: Make sure to evaluate the exponents before simplifying the expression.
Simplify the fractions: Simplify the fractions by finding a common denominator.
Add the fractions: Add the fractions by finding a common denominator.

Real-World Applications

Simplifying exponential expressions has many real-world applications, such as:

Science and engineering: Exponential expressions are used to model population growth, chemical reactions, and electrical circuits.
Finance: Exponential expressions are used to calculate interest rates, investment returns, and compound interest.
Computer science: Exponential expressions are used to model algorithms, data structures, and computational complexity.

Common Mistakes

Here are some common mistakes to avoid when simplifying exponential expressions:

Not evaluating the exponents: Make sure to evaluate the exponents before simplifying the expression.
Not simplifying the fractions: Simplify the fractions by finding a common denominator.
Not adding the fractions: Add the fractions by finding a common denominator.

Final Thoughts

Simplifying exponential expressions is a crucial skill that helps us evaluate complex equations and solve problems efficiently. By following the order of operations (PEMDAS) and using the rules of exponents, we can simplify exponential expressions and arrive at the final answer. Remember to evaluate the exponents, simplify the fractions, and add the fractions to get the correct answer.

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Q&A: Frequently Asked Questions

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 we have multiple operations in an expression. PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I simplify exponential expressions?

A: To simplify exponential expressions, you need to follow the order of operations (PEMDAS). First, evaluate the exponents, then simplify the fractions, and finally add the fractions.

Q: What is the rule for dividing exponential expressions with the same base?

A: When dividing two exponential expressions with the same base, you subtract the exponents. For example, $\frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2$ .

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that is common to both lists.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the number on top of a fraction, and the denominator is the number on the bottom of a fraction.

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.

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

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can list the factors of each number and find the largest number that is common to both lists.

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

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change.

Q: How do I simplify an expression with variables?

A: To simplify an expression with variables, you need to follow the order of operations (PEMDAS) and use the rules of exponents.

Q: What is the rule for multiplying exponential expressions with the same base?

A: When multiplying two exponential expressions with the same base, you add the exponents. For example, $2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32$ .

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

A: To simplify an expression with multiple operations, you need to follow the order of operations (PEMDAS) and use the rules of exponents.

Real-World Applications

Simplifying exponential expressions has many real-world applications, such as:

Science and engineering: Exponential expressions are used to model population growth, chemical reactions, and electrical circuits.
Finance: Exponential expressions are used to calculate interest rates, investment returns, and compound interest.
Computer science: Exponential expressions are used to model algorithms, data structures, and computational complexity.

Common Mistakes

Here are some common mistakes to avoid when simplifying exponential expressions:

Not evaluating the exponents: Make sure to evaluate the exponents before simplifying the expression.
Not simplifying the fractions: Simplify the fractions by finding a common denominator.
Not adding the fractions: Add the fractions by finding a common denominator.