A) (2^4 • 2^6) : (2^5 • 2^3)
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Introduction
In this article, we will explore the concept of exponentiation and how to simplify expressions involving powers of 2. We will use the given expression (2^4 • 2^6) : (2^5 • 2^3) as a case study to demonstrate the steps involved in simplifying complex expressions.
Understanding Exponentiation
Exponentiation is a mathematical operation that involves raising a number to a power. In the expression 2^4, for example, 2 is the base and 4 is the exponent. The result of 2^4 is 16, which is obtained by multiplying 2 by itself 4 times: 2 × 2 × 2 × 2 = 16.
Simplifying the Expression
To simplify the given expression, we need to follow the order of operations (PEMDAS):
- Evaluate the expressions inside the parentheses.
- Simplify the resulting expression.
Let's start by evaluating the expressions inside the parentheses:
(2^4 • 2^6) = 2^(4+6) = 2^10
(2^5 • 2^3) = 2^(5+3) = 2^8
Now, we can rewrite the original expression as:
(2^10) : (2^8)
Using the Quotient Rule
The quotient rule states that when dividing two powers with the same base, we subtract the exponents:
a^m ÷ a^n = a^(m-n)
In this case, we have:
(2^10) : (2^8) = 2^(10-8) = 2^2
Evaluating the Result
Now that we have simplified the expression, we can evaluate the result:
2^2 = 4
Therefore, the final answer is 4.
Conclusion
In this article, we have demonstrated how to simplify complex expressions involving powers of 2. We used the given expression (2^4 • 2^6) : (2^5 • 2^3) as a case study and applied the quotient rule to simplify the expression. The final answer is 4.
Frequently Asked Questions
Q: What is the order of operations?
A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Q: How do I simplify expressions involving powers of 2?
A: To simplify expressions involving powers of 2, you can use the quotient rule, which states that when dividing two powers with the same base, you subtract the exponents.
Q: What is the quotient rule?
A: The quotient rule states that when dividing two powers with the same base, you subtract the exponents: a^m ÷ a^n = a^(m-n).
Additional Resources
Related Topics
- Simplifying Expressions Involving Fractions
- Using the Quotient Rule with Fractions
- Simplifying Complex Expressions
Final Answer
The final answer is 4.
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Introduction
In this article, we will answer some frequently asked questions about exponentiation and simplifying expressions. We will cover topics such as the order of operations, simplifying expressions involving powers of 2, and using the quotient rule.
Q: What is the order of operations?
A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This means that when simplifying an expression, you should follow this order:
- Evaluate expressions inside parentheses.
- Evaluate any exponential expressions (such as 2^3).
- Perform any multiplication and division operations from left to right.
- Perform any addition and subtraction operations from left to right.
Q: How do I simplify expressions involving powers of 2?
A: To simplify expressions involving powers of 2, you can use the quotient rule, which states that when dividing two powers with the same base, you subtract the exponents. For example:
(2^4 • 2^6) : (2^5 • 2^3) = 2^(4+6) : 2^(5+3) = 2^10 : 2^8 = 2^(10-8) = 2^2
Q: What is the quotient rule?
A: The quotient rule states that when dividing two powers with the same base, you subtract the exponents: a^m ÷ a^n = a^(m-n).
Q: How do I simplify expressions involving fractions?
A: To simplify expressions involving fractions, you can use the following steps:
- Simplify the numerator and denominator separately.
- Use the quotient rule to simplify the expression.
For example:
(2/3) : (4/5) = (2/3) × (5/4) = 10/12 = 5/6
Q: What is the difference between a power and an exponent?
A: A power is the result of raising a number to a certain power. For example, 2^3 is a power. An exponent is the number that is being raised to a certain power. In the example above, 3 is the exponent.
Q: How do I simplify expressions involving negative exponents?
A: To simplify expressions involving negative exponents, you can use the following steps:
- Rewrite the expression with a positive exponent.
- Use the quotient rule to simplify the expression.
For example:
(2^(-3)) : (2^(-2)) = 2^(-3-(-2)) = 2^(-1) = 1/2
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. For example, x is a variable. A constant is a value that does not change. For example, 2 is a constant.
Q: How do I simplify expressions involving variables?
A: To simplify expressions involving variables, you can use the following steps:
- Simplify the expression using the order of operations.
- Use the quotient rule to simplify the expression.
For example:
(x^2) : (x^3) = x^(2-3) = x^(-1) = 1/x
Q: What is the difference between a rational expression and an irrational expression?
A: A rational expression is an expression that can be simplified to a fraction. For example, 2/3 is a rational expression. An irrational expression is an expression that cannot be simplified to a fraction. For example, √2 is an irrational expression.
Q: How do I simplify expressions involving rational and irrational expressions?
A: To simplify expressions involving rational and irrational expressions, you can use the following steps:
- Simplify the rational expression using the quotient rule.
- Simplify the irrational expression using the properties of radicals.
For example:
(2/3) : (√2) = (2/3) × (√2) = 2√2/3
Conclusion
In this article, we have answered some frequently asked questions about exponentiation and simplifying expressions. We have covered topics such as the order of operations, simplifying expressions involving powers of 2, and using the quotient rule. We hope that this article has been helpful in clarifying any confusion you may have had about these topics.
Frequently Asked Questions
Q: What is the order of operations?
A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Q: How do I simplify expressions involving powers of 2?
A: To simplify expressions involving powers of 2, you can use the quotient rule, which states that when dividing two powers with the same base, you subtract the exponents.
Q: What is the quotient rule?
A: The quotient rule states that when dividing two powers with the same base, you subtract the exponents: a^m ÷ a^n = a^(m-n).
Additional Resources
Related Topics
- Simplifying Expressions Involving Fractions
- Using the Quotient Rule with Fractions
- Simplifying Complex Expressions
Final Answer
The final answer is 4.