Find a simple expression for e{In x – 2ln y}

Mathematics is not only about numbers but it is about dealing with different calculations involving numbers and variables. This is what basically known as Algebra. Algebra is defined as the representation of calculations involving mathematical expressions that consist of numbers, operators, and variables. Numbers can be from 0 to 9, operators are the mathematical operators like +, -, ×, ÷, exponents, etc, variables like x, y, z, etc.

Exponents and Powers

Exponents and powers are the basic operators used in mathematical calculations, exponents are used to simplifying the complex calculations involving multiple self multiplications, self multiplications are basically numbers multiplied by themselves. For example, 7 × 7 × 7 × 7 × 7, can be simply written as 7⁵. Here, 7 is the base value and 5 is the exponent and the value is 16807. 11 × 11 × 11, can be written as 11³, here, 11 is the base value and 3 is the exponent or power of 11. The value of 11³ is 1331.

Exponent is defined as the power given to a number, the number of times it is multiplied by itself. If an expression is written as cx^y where c is a constant, c will be the coefficient, x is the base and y is the exponent. If a number say p, is multiplied n times, n will be the exponent of p. It will be written as

p × p × p × p … n times = pⁿ

Basic rules of Exponents

There are certain basic rules defined for exponents in order to solve the exponential expressions along with the other mathematical operations, for example, if there are the product of two exponents, it can be simplified to make the calculation easier and is known as product rule, let’s look at some of the basic rules of exponents,

• Product Rule ⇢ aⁿ × a^m = a^{n + m}
• Quotient Rule ⇢ aⁿ / a^m = a^{n – m}
• Power Rule ⇢ (aⁿ)^m = a^{n × m} or ^m√aⁿ = a^n/m
• Negative Exponent Rule ⇢ a^-m = 1/a^m
• Zero Rule ⇢ a⁰ = 1
• One Rule ⇢ a¹ = a

Basic Rules of Logarithm

In mathematics, logarithms are defined as the inverse of the exponents. Logarithms count the number of presence of the same factor in repeated multiplication. For instance, 10000 = 10 × 10 × 10 × 10 = 10⁴, here in this case, the logarithm base 10 of 10000 is 4. Following are some of the most basic rules of logarithms,

• Product Rule ⇢ log_b mn = log_b m + log_b n
• Quotient Rule ⇢ log_b m/n = log_b m – log_b n
• Power Rule ⇢ log_b m^p = plog_b m
• Equality Rule ⇢ If log_b m = log_b n. Then, m = n

Find a simple expression for e^{{In x – 2ln y}}

Solution:

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules and basic logarithmic rules, looking at the expression e^{{In x-2ln y}}, it is observed that the power rule of logarithms can be easily applied to this expression,

Power Rule ⇢ log_b m^p = plog_b m

e^{{In x-2ln y}} = e^{{In x – ln y2}}

Apply the quotient rule of exponents.

Quotient Rule ⇢ aⁿ / a^m = a^{n – m}

e^{{In x-ln y2}} = e^lnx / e^lny2

= x / y²

Therefore, x / y² is the value obtained.

Similar Problems

Question 1: What is 10m⁵(3m⁷)?

Solution:

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules, looking at the expression 10m⁵(3m⁷), it is observed that the product rule of exponents can be easily applied to this expression,

Step 1: Remove the parenthesis and write terms with their exponents.

10m⁵(3m⁷) = 10m⁵ × 3m⁷

Step 2: Apply the product rule of exponents.

Product Rule ⇢ aⁿ × a^m = a^{n + m}

10m⁵ × 3m⁷ = 10 × 3m^{(5 + 7)}

Therefore, 30m¹² is the value obtained.

Question 2: Simplify (x¹⁰)(x²)

Solution:

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules, looking at the expression (x¹⁰)(x²), it is observed that the product rule of exponents can be easily applied to this expression,

Product Rule ⇢ aⁿ × a^m = a^{n + m}

x¹⁰ × x² = x^{(10 + 2)}

= x¹²

Therefore, x¹² is the value obtained.

Question 3: Simplify e^{{6In x – 5ln y}}

Solution:

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules and basic logarithmic rules, looking at the expression e^{{6In x-5ln y}}, it is observed that the power rule of logarithms can be easily applied to this expression,

Power Rule ⇢ log_b m^p = plog_b m

e^{{6In x – 5ln y}} = e^{{In x6 – ln y5}}

Apply the quotient rule of exponents.

Quotient Rule ⇢ aⁿ / a^m = a^{n – m}

e^{{In x6 – ln y5}} = elnx⁶ / elny²

= x⁶ / y⁵

Therefore, x⁶ / y⁵ is the value obtained.