Simplifying Exponential Expressions Using The Laws Of Exponents

In mathematics, especially in algebra and calculus, simplifying expressions involving exponents is a fundamental skill. The laws of exponents provide a set of rules that allow us to manipulate and simplify expressions containing powers. This article focuses on using these laws to simplify a specific exponential expression. We will delve into the laws of exponents, explain each rule with examples, and then apply these rules step-by-step to simplify the given expression: $ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} $. By the end of this article, you will have a clear understanding of how to simplify expressions with rational exponents using the power of a power rule and be able to apply these techniques to similar problems.

Understanding the Laws of Exponents

To effectively simplify exponential expressions, it is crucial to understand the fundamental laws of exponents. These laws provide the framework for manipulating and reducing complex expressions into simpler forms. Mastery of these laws is essential for success in algebra, calculus, and various other branches of mathematics. Let's explore some of the key laws of exponents with detailed explanations and examples.

1. Product of Powers Rule

The product of powers rule states that when multiplying two exponential expressions with the same base, you can add the exponents. Mathematically, this is expressed as:

$a^m \cdot a^n = a^{m+n}$

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

Example:

Consider the expression $2^3 \cdot 2^4$. Here, the base is 2, and the exponents are 3 and 4. Applying the product of powers rule:

$2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

This rule simplifies the multiplication of exponential expressions by combining the exponents, making the calculation more straightforward. For instance, this is particularly useful when dealing with large exponents or variables in exponents, as it reduces the complexity of the expression.

2. Quotient of Powers Rule

The quotient of powers rule applies when dividing two exponential expressions with the same base. It states that you can subtract the exponent in the denominator from the exponent in the numerator. The rule is expressed as:

$\frac{a^m}{a^n} = a^{m-n}$

Where a is the base, and m and n are the exponents. It is important to note that a cannot be zero, as division by zero is undefined.

Example:

Consider the expression $\frac{5^6}{5^2}$. Here, the base is 5, and the exponents are 6 and 2. Applying the quotient of powers rule:

$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$

This rule helps simplify division problems involving exponents by reducing the exponents to a single term. This is especially helpful in algebraic manipulations and simplifying complex fractions involving exponential terms.

3. Power of a Power Rule

The power of a power rule is crucial when raising an exponential expression to another power. It states that you can multiply the exponents. The rule is expressed as:

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

Where a is the base, and m and n are the exponents.

Example:

Consider the expression $(3^2)^3$. Here, the base is 3, and the exponents are 2 and 3. Applying the power of a power rule:

$(3^2)^3 = 3^{2 \cdot 3} = 3^6 = 729$

This rule is particularly useful when dealing with nested exponents, as it simplifies the expression into a single exponent. This simplification is widely used in various mathematical contexts, including polynomial manipulations and calculus.

4. Power of a Product Rule

The power of a product rule is used when raising a product to a power. It states that you can distribute the exponent to each factor within the product. The rule is expressed as:

$(ab)^n = a^n b^n$

Where a and b are the bases, and n is the exponent.

Example:

Consider the expression $(2x)^3$. Here, the product is 2x, and the exponent is 3. Applying the power of a product rule:

$(2x)^3 = 2^3 x^3 = 8x^3$

This rule is essential for expanding expressions and is commonly used in algebraic simplifications and polynomial expansions. It allows for the distribution of exponents across products, making it easier to handle complex expressions.

5. Power of a Quotient Rule

The power of a quotient rule is used when raising a quotient to a power. It states that you can distribute the exponent to both the numerator and the denominator. The rule is expressed as:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Where a and b are the bases, n is the exponent, and b cannot be zero.

Example:

Consider the expression $\left(\frac{4}{y}\right)^2$. Here, the quotient is $\frac{4}{y}$, and the exponent is 2. Applying the power of a quotient rule:

$\left(\frac{4}{y}\right)^2 = \frac{4^2}{y^2} = \frac{16}{y^2}$

This rule is particularly useful in simplifying fractions with exponents, as it allows the exponent to be applied separately to both the numerator and the denominator, simplifying complex rational expressions.

6. Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. The rule is expressed as:

$a^0 = 1$

Where a is any non-zero number.

Example:

Consider the expression $7^0$. Applying the zero exponent rule:

$7^0 = 1$

This rule is a fundamental concept in exponent manipulation and is used extensively in simplifying expressions. It provides a consistent and straightforward way to handle zero exponents in calculations.

7. Negative Exponent Rule

The negative exponent rule states that a number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent. The rule is expressed as:

$a^{-n} = \frac{1}{a^n}$

Where a is any non-zero number and n is the exponent.

Example:

Consider the expression $3^{-2}$. Applying the negative exponent rule:

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

This rule is essential for converting expressions with negative exponents into fractions with positive exponents, making it easier to simplify and evaluate expressions. It is widely used in algebraic manipulations and calculus.

Step-by-Step Simplification of $ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} $

Now that we have a solid understanding of the laws of exponents, let's apply these principles to simplify the given expression: $ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} $. This expression involves raising an exponential term to another power, which is a classic application of the power of a power rule.

Step 1: Identify the Rule to Apply

The expression $ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} $ is in the form of $(a^m)^n$, where:

• a = 6
• m = $\frac{5}{6}$
• n = $\frac{2}{11}$

This clearly indicates that we need to apply the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

Step 2: Apply the Power of a Power Rule

Using the power of a power rule, we multiply the exponents:

$ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} = 6^{\frac{5}{6} \cdot \frac{2}{11}} $

Step 3: Multiply the Exponents

Now, we multiply the fractions in the exponent:

$\frac{5}{6} \cdot \frac{2}{11} = \frac{5 \cdot 2}{6 \cdot 11} = \frac{10}{66}$

Step 4: Simplify the Fraction

Next, we simplify the fraction $\frac{10}{66}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$\frac{10}{66} = \frac{10 \div 2}{66 \div 2} = \frac{5}{33}$

Step 5: Substitute the Simplified Exponent

Now, we substitute the simplified exponent back into the expression:

$6^{\frac{5}{6} \cdot \frac{2}{11}} = 6^{\frac{5}{33}}$

Step 6: Final Simplified Expression

Therefore, the simplified form of the given expression is:

$ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} = 6^{\frac{5}{33}} $

This is the final simplified form, as the exponent $\frac{5}{33}$ cannot be further reduced, and the base 6 is a prime number raised to a rational exponent.

Additional Examples

To further solidify your understanding of simplifying expressions with the laws of exponents, let's go through some additional examples. These examples will cover different scenarios and variations, providing a broader perspective on how to apply the rules effectively. Each example will be broken down step-by-step to ensure clarity and comprehension.

Example 1: Simplifying with Multiple Rules

Consider the expression: $\frac{(2^2 \cdot 3^3)^2}{2^3 \cdot 3^4}$

This example combines several laws of exponents, including the power of a product rule, the power of a power rule, and the quotient of powers rule. Let’s simplify it step-by-step.

Step 1: Apply the Power of a Product Rule

First, apply the power of a product rule to the numerator:

$(2^2 \cdot 3^3)^2 = (2^2)^2 \cdot (3^3)^2$

Step 2: Apply the Power of a Power Rule

Next, apply the power of a power rule to both terms in the numerator:

$(2^2)^2 = 2^{2 \cdot 2} = 2^4$

$(3^3)^2 = 3^{3 \cdot 2} = 3^6$

So, the numerator becomes:

$2^4 \cdot 3^6$

Step 3: Rewrite the Expression

Now, rewrite the entire expression:

$\frac{2^4 \cdot 3^6}{2^3 \cdot 3^4}$

Step 4: Apply the Quotient of Powers Rule

Apply the quotient of powers rule to both the base 2 and the base 3:

$\frac{2^4}{2^3} = 2^{4-3} = 2^1 = 2$

$\frac{3^6}{3^4} = 3^{6-4} = 3^2 = 9$

Step 5: Final Simplification

Multiply the simplified terms:

$2 \cdot 9 = 18$

Therefore, the simplified expression is:

$\frac{(2^2 \cdot 3^3)^2}{2^3 \cdot 3^4} = 18$

Example 2: Simplifying with Negative Exponents

Consider the expression: $\frac{5^{-2} \cdot 10^3}{2^{-3} \cdot 5^4}$

This example involves negative exponents and requires the application of the negative exponent rule along with other exponent laws.

Step 1: Apply the Negative Exponent Rule

First, apply the negative exponent rule to the terms with negative exponents:

$5^{-2} = \frac{1}{5^2}$

$2^{-3} = \frac{1}{2^3}$

Step 2: Rewrite the Expression

Rewrite the expression with positive exponents:

$\frac{\frac{1}{5^2} \cdot 10^3}{\frac{1}{2^3} \cdot 5^4}$

Step 3: Simplify the Fraction

To simplify the fraction, multiply the numerator and denominator by $5^2 \cdot 2^3$ to eliminate the fractions in the numerator and denominator:

$\frac{\frac{1}{5^2} \cdot 10^3}{\frac{1}{2^3} \cdot 5^4} \cdot \frac{5^2 \cdot 2^3}{5^2 \cdot 2^3} = \frac{10^3 \cdot 2^3}{5^4 \cdot 5^2}$

Step 4: Express 10 as a Product of 2 and 5

Express 10 as $2 \cdot 5$:

$10^3 = (2 \cdot 5)^3 = 2^3 \cdot 5^3$

Step 5: Substitute and Simplify

Substitute $10^3$ back into the expression:

$\frac{2^3 \cdot 5^3 \cdot 2^3}{5^4 \cdot 5^2}$

Combine like terms:

$\frac{2^{3+3} \cdot 5^3}{5^{4+2}} = \frac{2^6 \cdot 5^3}{5^6}$

Step 6: Apply the Quotient of Powers Rule

Apply the quotient of powers rule to the base 5:

$\frac{5^3}{5^6} = 5^{3-6} = 5^{-3} = \frac{1}{5^3}$

Step 7: Final Simplification

So, the expression simplifies to:

$2^6 \cdot \frac{1}{5^3} = \frac{2^6}{5^3} = \frac{64}{125}$

Therefore, the simplified expression is:

$\frac{5^{-2} \cdot 10^3}{2^{-3} \cdot 5^4} = \frac{64}{125}$

Example 3: Simplifying with Rational Exponents

Consider the expression: $(8^{\frac{2}{3}} \cdot 4^{\frac{1}{2}})^2$

This example involves rational exponents and the application of the power of a product rule and the power of a power rule.

Step 1: Apply the Power of a Product Rule

First, apply the power of a product rule:

$(8^{\frac{2}{3}} \cdot 4^{\frac{1}{2}})^2 = (8^{\frac{2}{3}})^2 \cdot (4^{\frac{1}{2}})^2$

Step 2: Apply the Power of a Power Rule

Next, apply the power of a power rule to both terms:

$(8^{\frac{2}{3}})^2 = 8^{\frac{2}{3} \cdot 2} = 8^{\frac{4}{3}}$

$(4^{\frac{1}{2}})^2 = 4^{\frac{1}{2} \cdot 2} = 4^1 = 4$

Step 3: Rewrite the Expression

Rewrite the expression:

$8^{\frac{4}{3}} \cdot 4$

Step 4: Express 8 and 4 as Powers of 2

Express 8 and 4 as powers of 2:

$8 = 2^3$

$4 = 2^2$

Step 5: Substitute and Simplify

Substitute these values back into the expression:

$(2^3)^{\frac{4}{3}} \cdot 2^2$

Apply the power of a power rule:

$2^{3 \cdot \frac{4}{3}} \cdot 2^2 = 2^4 \cdot 2^2$

Step 6: Apply the Product of Powers Rule

Apply the product of powers rule:

$2^{4+2} = 2^6$

Step 7: Final Simplification

Calculate $2^6$:

$2^6 = 64$

Therefore, the simplified expression is:

$(8^{\frac{2}{3}} \cdot 4^{\frac{1}{2}})^2 = 64$

Common Mistakes to Avoid

When simplifying expressions with exponents, it's easy to make mistakes if you're not careful. Understanding common pitfalls can help you avoid errors and improve your accuracy. Let’s discuss some frequent mistakes and how to prevent them.

1. Incorrect Application of the Product of Powers Rule

Mistake: Adding bases instead of exponents.

Correct Rule: $a^m \cdot a^n = a^{m+n}$

Incorrect Example: $2^3 \cdot 2^4 = 4^{3+4} = 4^7$ (Incorrect)

Correct Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$ (Correct)

Prevention: Always remember that the product of powers rule applies only to the exponents when the bases are the same. Do not add the bases; instead, add the exponents.

2. Incorrect Application of the Quotient of Powers Rule

Mistake: Subtracting the base or exponents in the wrong order.

Correct Rule: $\frac{a^m}{a^n} = a^{m-n}$

Incorrect Example: $\frac{3^5}{3^2} = 3^{2-5} = 3^{-3}$ (Incorrect, though the final result can be made correct with proper handling of the negative exponent in the next step.)

Correct Example: $\frac{3^5}{3^2} = 3^{5-2} = 3^3$ (Correct)

Prevention: Always subtract the exponent in the denominator from the exponent in the numerator. Ensure you maintain the correct order of subtraction.

3. Incorrect Application of the Power of a Power Rule

Mistake: Adding exponents instead of multiplying them.

Correct Rule: $(a^m)^n = a^{m \cdot n}$

Incorrect Example: $(4^2)^3 = 4^{2+3} = 4^5$ (Incorrect)

Correct Example: $(4^2)^3 = 4^{2 \cdot 3} = 4^6$ (Correct)

Prevention: Remember that when raising a power to a power, you multiply the exponents, not add them.

4. Incorrect Application of the Power of a Product or Quotient Rule

Mistake: Applying the exponent to only one factor or term within the parentheses.

Correct Rule:

• $(ab)^n = a^n b^n$
• $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Incorrect Example: $(2x)^3 = 2x^3$ (Incorrect)

Correct Example: $(2x)^3 = 2^3 x^3 = 8x^3$ (Correct)

Prevention: Always distribute the exponent to each factor within the parentheses for products and to both the numerator and the denominator for quotients.

5. Mistakes with Negative Exponents

Mistake: Treating negative exponents as negative numbers.

Correct Rule: $a^{-n} = \frac{1}{a^n}$

Incorrect Example: $5^{-2} = -5^2 = -25$ (Incorrect)

Correct Example: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$ (Correct)

Prevention: Understand that a negative exponent indicates a reciprocal. The base with the negative exponent should be moved to the denominator (or numerator if it’s already in the denominator), and the exponent becomes positive.

6. Mistakes with the Zero Exponent Rule

Mistake: Assuming $a^0 = 0$ instead of 1.

Correct Rule: $a^0 = 1$ (for $a \neq 0$)

Incorrect Example: $7^0 = 0$ (Incorrect)

Correct Example: $7^0 = 1$ (Correct)

Prevention: Any non-zero number raised to the power of zero is 1. Remember this rule to avoid confusion.

7. Forgetting to Simplify Fractions

Mistake: Leaving exponents as fractions that can be simplified.

Example: Leaving an answer as $6^{\frac{10}{66}}$ without simplifying to $6^{\frac{5}{33}}$

Prevention: Always simplify fractions in exponents to their lowest terms. This ensures the expression is in its simplest form.

8. Combining Terms Incorrectly

Mistake: Combining terms that do not have the same base or exponent.

Example: $2^3 + 2^2 = 2^{3+2} = 2^5$ (Incorrect)

Prevention: You can only add or subtract terms if they are like terms (i.e., they have the same base and exponent). In the incorrect example, you would need to calculate each term separately: $2^3 = 8$ and $2^2 = 4$, so $8 + 4 = 12$.

By being aware of these common mistakes and practicing the correct application of the exponent rules, you can significantly improve your accuracy and confidence in simplifying exponential expressions.

Conclusion

In this article, we explored the fundamental laws of exponents and applied them to simplify the expression $ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} $. We began by outlining the key laws of exponents, including the product of powers, quotient of powers, power of a power, power of a product, power of a quotient, zero exponent, and negative exponent rules. Each rule was explained with detailed examples to ensure a clear understanding. We then focused on simplifying the given expression, demonstrating a step-by-step process using the power of a power rule. Additionally, we worked through several examples to illustrate the application of these rules in various scenarios, including expressions with multiple rules, negative exponents, and rational exponents. Finally, we addressed common mistakes to avoid when simplifying exponential expressions, providing practical tips to enhance accuracy. By mastering these concepts and techniques, you can confidently simplify a wide range of exponential expressions in mathematics and related fields. The simplified form of the expression $ \left(6^{{\frac{5}{6}}\right)}{\frac{2}{11}} $ is $6^{\frac{5}{33}}$, which exemplifies the power of exponent rules in reducing complex expressions to their simplest forms.