Simplify The Expression: ${ \frac{5^{\frac{1}{2}} \times 20^{\frac{1}{4}} \times 2 {\frac{3}{2}}}{5 {\frac{3}{4}}} }$

Introduction

In mathematics, exponents and indices are fundamental concepts that play a crucial role in simplifying complex expressions. When dealing with expressions involving exponents and indices, it's essential to understand the rules and properties that govern their behavior. In this article, we will delve into the world of exponents and indices, exploring the rules and techniques for simplifying expressions involving these mathematical concepts.

Understanding Exponents and Indices

Exponents and indices are used to represent repeated multiplication of a number. An exponent is a small number that is raised to a power, indicating how many times the base number is multiplied by itself. For example, in the expression $2^3$, the exponent 3 indicates that the base number 2 is multiplied by itself three times, resulting in $2 \times 2 \times 2 = 8$. Indices, on the other hand, are used to represent the power to which a number is raised. For instance, in the expression $2^3$, the index 3 represents the power to which the base number 2 is raised.

Simplifying Expressions with Exponents and Indices

When simplifying expressions involving exponents and indices, it's essential to apply the rules and properties of exponents and indices. One of the fundamental rules is the product rule, which states that when multiplying two numbers with the same base, we add their exponents. For example, in the expression $2^3 \times 2^4$, we can simplify it by adding the exponents, resulting in $2^{3+4} = 2^7$. Another important rule is the quotient rule, which states that when dividing two numbers with the same base, we subtract their exponents. For instance, in the expression $\frac{2^3}{2^4}$, we can simplify it by subtracting the exponents, resulting in $2^{3-4} = 2^{-1}$.

Applying the Rules of Exponents and Indices

Now that we have a solid understanding of the rules and properties of exponents and indices, let's apply them to simplify the given expression: $\frac{5^{\frac{1}{2}} \times 20^{\frac{1}{4}} \times 2^{\frac{3}{2}}}{5^{\frac{3}{4}}}$. To simplify this expression, we need to apply the rules of exponents and indices, taking into account the product and quotient rules.

Step 1: Simplify the Numerator

The numerator of the expression is $5^{\frac{1}{2}} \times 20^{\frac{1}{4}} \times 2^{\frac{3}{2}}$. To simplify this expression, we need to apply the product rule, which states that when multiplying two numbers with the same base, we add their exponents. However, in this case, we have three different bases, so we need to apply the product rule separately to each pair of numbers.

Step 1.1: Simplify the First Pair of Numbers

The first pair of numbers is $5^{\frac{1}{2}}$ and $20^{\frac{1}{4}}$. To simplify this expression, we need to apply the product rule, which states that when multiplying two numbers with the same base, we add their exponents. However, in this case, we have two different bases, so we need to apply the product rule separately to each number.

Step 1.1.1: Simplify the First Number

The first number is $5^{\frac{1}{2}}$. To simplify this expression, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents. However, in this case, we have a single exponent, so we don't need to apply this rule.

Step 1.1.2: Simplify the Second Number

The second number is $20^{\frac{1}{4}}$. To simplify this expression, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents. However, in this case, we have a single exponent, so we don't need to apply this rule.

Step 1.2: Simplify the Second Pair of Numbers

The second pair of numbers is $20^{\frac{1}{4}}$ and $2^{\frac{3}{2}}$. To simplify this expression, we need to apply the product rule, which states that when multiplying two numbers with the same base, we add their exponents. However, in this case, we have two different bases, so we need to apply the product rule separately to each number.

Step 1.2.1: Simplify the First Number

The first number is $20^{\frac{1}{4}}$. To simplify this expression, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents. However, in this case, we have a single exponent, so we don't need to apply this rule.

Step 1.2.2: Simplify the Second Number

The second number is $2^{\frac{3}{2}}$. To simplify this expression, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents. However, in this case, we have a single exponent, so we don't need to apply this rule.

Step 2: Simplify the Denominator

The denominator of the expression is $5^{\frac{3}{4}}$. To simplify this expression, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents. However, in this case, we have a single exponent, so we don't need to apply this rule.

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by applying the quotient rule, which states that when dividing two numbers with the same base, we subtract their exponents.

Step 3.1: Simplify the Numerator

The numerator of the expression is $5^{\frac{1}{2}} \times 20^{\frac{1}{4}} \times 2^{\frac{3}{2}}$. To simplify this expression, we need to apply the product rule, which states that when multiplying two numbers with the same base, we add their exponents.

Step 3.1.1: Simplify the First Pair of Numbers

The first pair of numbers is $5^{\frac{1}{2}}$ and $20^{\frac{1}{4}}$. To simplify this expression, we need to apply the product rule, which states that when multiplying two numbers with the same base, we add their exponents.

Step 3.1.2: Simplify the Second Pair of Numbers

The second pair of numbers is $20^{\frac{1}{4}}$ and $2^{\frac{3}{2}}$. To simplify this expression, we need to apply the product rule, which states that when multiplying two numbers with the same base, we add their exponents.

Step 3.2: Simplify the Denominator

The denominator of the expression is $5^{\frac{3}{4}}$. To simplify this expression, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents.

Step 3.3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by applying the quotient rule, which states that when dividing two numbers with the same base, we subtract their exponents.

Step 4: Final Simplification

After applying the quotient rule, we get the final simplified expression: $5^{\frac{1}{2} - \frac{3}{4}} \times 2^{\frac{3}{2}}$. To simplify this expression further, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents.

Step 4.1: Simplify the Exponent

The exponent is $\frac{1}{2} - \frac{3}{4}$. To simplify this expression, we need to find a common denominator, which is 4.

Step 4.1.1: Simplify the Exponent

The exponent is $\frac{1}{2} - \frac{3}{4}$. To simplify this expression, we need to convert the fraction $\frac{1}{2}$ to have a denominator of 4.

Step 4.1.2: Simplify the Exponent

The exponent is $\frac{1}{2} - \frac{3}{4}$. To simplify this expression, we need to subtract the fractions.

Step 4.2: Simplify the Expression

Now that we have simplified the exponent, we can simplify the expression by applying the rule of exponents, which states that when raising a power to a power, we multiply the exponents.

Step 4.2.1: Simplify the Expression

The expression is $5^{\frac{1}{2} - \frac{3}{4}} \times 2^{\frac{3}{2}}$. To simplify this expression, we need to apply the rule of exponents, which states that when raising a power to a power, we multiply the exponents.

Step 4.2.2: Simplify the Expression

The expression is $5^{\frac{1}{2}

Introduction

In our previous article, we explored the world of exponents and indices, delving into the rules and techniques for simplifying expressions involving these mathematical concepts. We applied the product and quotient rules to simplify the given expression: $\frac{5^{\frac{1}{2}} \times 20^{\frac{1}{4}} \times 2^{\frac{3}{2}}}{5^{\frac{3}{4}}}$. In this article, we will answer some of the most frequently asked questions related to simplifying expressions with exponents and indices.

Q&A

Q: What is the difference between an exponent and an index?

A: An exponent is a small number that is raised to a power, indicating how many times the base number is multiplied by itself. An index, on the other hand, is used to represent the power to which a number is raised.

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

A: To simplify an expression with multiple exponents, you need to apply the product rule, which states that when multiplying two numbers with the same base, you add their exponents. You also need to apply the quotient rule, which states that when dividing two numbers with the same base, you subtract their exponents.

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

A: When multiplying numbers with the same base, you add their exponents. For example, in the expression $2^3 \times 2^4$, you can simplify it by adding the exponents, resulting in $2^{3+4} = 2^7$.

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

A: When dividing numbers with the same base, you subtract their exponents. For instance, in the expression $\frac{2^3}{2^4}$, you can simplify it by subtracting the exponents, resulting in $2^{3-4} = 2^{-1}$.

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

A: To simplify an expression with a negative exponent, you need to apply the rule of negative exponents, which states that a negative exponent is equal to the reciprocal of the positive exponent. For example, in the expression $2^{-3}$, you can simplify it by taking the reciprocal of the positive exponent, resulting in $\frac{1}{2^3}$.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, you multiply the exponents. For example, in the expression $(2^3)^4$, you can simplify it by multiplying the exponents, resulting in $2^{3 \times 4} = 2^{12}$.

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

A: To simplify an expression with multiple bases, you need to apply the product rule, which states that when multiplying two numbers with the same base, you add their exponents. You also need to apply the quotient rule, which states that when dividing two numbers with the same base, you subtract their exponents.

Q: What is the rule for simplifying an expression with a fraction as an exponent?

A: When simplifying an expression with a fraction as an exponent, you need to apply the rule of fractions as exponents, which states that a fraction as an exponent is equal to the base raised to the power of the numerator divided by the denominator. For example, in the expression $2^{\frac{1}{2}}$, you can simplify it by taking the square root of the base, resulting in $\sqrt{2}$.

Conclusion

Simplifying expressions with exponents and indices can be a challenging task, but with the right rules and techniques, it can be made easier. By applying the product and quotient rules, you can simplify expressions with multiple exponents and bases. Remember to always follow the rules of exponents and indices, and you will be able to simplify even the most complex expressions.

Final Tips

• Always start by simplifying the numerator and denominator separately.
• Use the product rule to simplify expressions with multiple bases.
• Use the quotient rule to simplify expressions with multiple bases.
• Remember to apply the rule of negative exponents when simplifying expressions with negative exponents.
• Use the rule of fractions as exponents to simplify expressions with fractions as exponents.

By following these tips and applying the rules of exponents and indices, you will be able to simplify even the most complex expressions and become a master of algebra.