Use The Product Of Powers And The Quotient Of Powers To Combine The Powers Of 2. Then Evaluate The Expression Using The Order Of Operations.$\[ 5 \cdot 2^{\frac{1}{2}} \cdot 2^{-\frac{1}{2}} - \frac{5 \cdot 2^{\frac{1}{2}}}{2^{\frac{1}{2}}} =

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Introduction

In mathematics, the product of powers and quotient of powers are two fundamental rules used to simplify expressions involving exponents. These rules enable us to combine powers of the same base, making it easier to evaluate complex expressions. In this article, we will explore how to use the product of powers and quotient of powers to combine the powers of 2, and then evaluate the resulting expression using the order of operations.

The Product of Powers Rule

The product of powers rule states that when multiplying two or more powers with the same base, we can add the exponents. Mathematically, this can be represented as:

a^m * a^n = a^(m+n)

where a is the base and m and n are the exponents.

The Quotient of Powers Rule

The quotient of powers rule states that when dividing two powers with the same base, we can subtract the exponents. Mathematically, this can be represented as:

a^m / a^n = a^(m-n)

where a is the base and m and n are the exponents.

Combining Powers of 2

Using the product of powers and quotient of powers rules, we can simplify the given expression:

5⋅212⋅2−12−5⋅212212{ 5 \cdot 2^{\frac{1}{2}} \cdot 2^{-\frac{1}{2}} - \frac{5 \cdot 2^{\frac{1}{2}}}{2^{\frac{1}{2}}} }

First, let's simplify the first term using the product of powers rule:

5⋅212⋅2−12=5⋅212−12{ 5 \cdot 2^{\frac{1}{2}} \cdot 2^{-\frac{1}{2}} = 5 \cdot 2^{\frac{1}{2} - \frac{1}{2}} }

Using the quotient of powers rule, we can simplify the exponent:

212−12=20{ 2^{\frac{1}{2} - \frac{1}{2}} = 2^0 }

Since any number raised to the power of 0 is equal to 1, we can simplify the first term:

5⋅20=5⋅1=5{ 5 \cdot 2^0 = 5 \cdot 1 = 5 }

Now, let's simplify the second term using the quotient of powers rule:

5⋅212212=5⋅212−12{ \frac{5 \cdot 2^{\frac{1}{2}}}{2^{\frac{1}{2}}} = 5 \cdot 2^{\frac{1}{2} - \frac{1}{2}} }

Using the quotient of powers rule, we can simplify the exponent:

212−12=20{ 2^{\frac{1}{2} - \frac{1}{2}} = 2^0 }

Since any number raised to the power of 0 is equal to 1, we can simplify the second term:

5⋅20=5⋅1=5{ 5 \cdot 2^0 = 5 \cdot 1 = 5 }

Now that we have simplified both terms, we can evaluate the expression:

5−5=0{ 5 - 5 = 0 }

Therefore, the final answer is 0.

Conclusion

In this article, we used the product of powers and quotient of powers rules to combine the powers of 2 in the given expression. We simplified the expression using these rules and then evaluated the resulting expression using the order of operations. The final answer is 0.

Order of Operations

The order of operations is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is as follows:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following the order of operations, we can ensure that mathematical expressions are evaluated correctly and consistently.

Example Problems

Here are a few example problems that demonstrate the use of the product of powers and quotient of powers rules:

  • Example 1: Simplify the expression 2^3 * 2^2 using the product of powers rule.
    • Using the product of powers rule, we can add the exponents:
      • 2^3 * 2^2 = 2^(3+2)
      • 2^(3+2) = 2^5
      • Therefore, the simplified expression is 2^5.
  • Example 2: Simplify the expression 2^4 / 2^2 using the quotient of powers rule.
    • Using the quotient of powers rule, we can subtract the exponents:
      • 2^4 / 2^2 = 2^(4-2)
      • 2^(4-2) = 2^2
      • Therefore, the simplified expression is 2^2.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two or more powers with the same base, we can add the exponents. Mathematically, this can be represented as:

a^m * a^n = a^(m+n)

where a is the base and m and n are the exponents.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, we can subtract the exponents. Mathematically, this can be represented as:

a^m / a^n = a^(m-n)

where a is the base and m and n are the exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents of the two powers with the same base. For example:

2^3 * 2^2 = 2^(3+2) = 2^5

Q: How do I apply the quotient of powers rule?

A: To apply the quotient of powers rule, simply subtract the exponents of the two powers with the same base. For example:

2^4 / 2^2 = 2^(4-2) = 2^2

Q: What if I have a power with a negative exponent?

A: If you have a power with a negative exponent, you can rewrite it as a fraction with a positive exponent. For example:

2^(-3) = 1/2^3

Q: Can I apply the product of powers rule to more than two powers?

A: Yes, you can apply the product of powers rule to more than two powers. For example:

2^3 * 2^2 * 2^1 = 2^(3+2+1) = 2^6

Q: Can I apply the quotient of powers rule to more than two powers?

A: Yes, you can apply the quotient of powers rule to more than two powers. For example:

2^4 / 2^2 / 2^1 = 2^(4-2-1) = 2^1

Q: How do I evaluate an expression with multiple powers?

A: To evaluate an expression with multiple powers, follow the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following the order of operations, you can ensure that mathematical expressions are evaluated correctly and consistently.

Common Mistakes to Avoid

  • Not following the order of operations: Make sure to follow the order of operations when evaluating expressions with multiple powers.
  • Not simplifying powers with negative exponents: Make sure to simplify powers with negative exponents by rewriting them as fractions with positive exponents.
  • Not applying the product of powers rule correctly: Make sure to add the exponents when multiplying powers with the same base.
  • Not applying the quotient of powers rule correctly: Make sure to subtract the exponents when dividing powers with the same base.

By avoiding these common mistakes, you can ensure that you are applying the product of powers and quotient of powers rules correctly and consistently.