Problem 6
Question
In \(3-10,\) write each expression as a rational number without an exponent. $$ \left(\frac{1}{2}\right)^{-1} $$
Step-by-Step Solution
Verified Answer
The expression \(\left(\frac{1}{2}\right)^{-1}\) is equal to 2.
1Step 1: Understand Negative Exponents
The expression \(\left(\frac{1}{2}\right)^{-1}\) involves a negative exponent. A negative exponent \(n^{-m}\) indicates that we take the reciprocal of the base and use the positive exponent: \(n^{-m} = \frac{1}{n^m}\).
2Step 2: Apply the Rule for Negative Exponents
Using the rule for negative exponents, write \(\left(\frac{1}{2}\right)^{-1}\) as the reciprocal of \(\frac{1}{2}\) raised to the power of 1. This gives us \(\frac{2}{1}\).
3Step 3: Simplify the Expression
Simplify \(\frac{2}{1}\) to its simplest form, which is just 2. Therefore, the expression \(\left(\frac{1}{2}\right)^{-1}\) simplifies to 2 without any exponents.
Key Concepts
ReciprocalSimplifying ExpressionsRational Numbers
Reciprocal
In mathematics, the concept of the reciprocal is fundamental when dealing with fractions and negative exponents.
The reciprocal of any number is essentially what you multiply the original number by to get the product of 1.
For instance, the reciprocal of a fraction like \( \frac{a}{b} \) is simply \( \frac{b}{a} \).
Let’s say you have the fraction \( \frac{1}{2} \). Its reciprocal would be \( \frac{2}{1} \) or just 2.
This concept arises naturally when handling expressions with negative exponents.
A negative exponent prompts you to take the reciprocal of the base.
This can be expressed generally as follows: if you have a number \( a^{-n} \), its equivalent will be \( \frac{1}{a^{n}} \).
Taking the reciprocal shifts the exponent from negative to positive, making it easier to compute.
The reciprocal of any number is essentially what you multiply the original number by to get the product of 1.
For instance, the reciprocal of a fraction like \( \frac{a}{b} \) is simply \( \frac{b}{a} \).
Let’s say you have the fraction \( \frac{1}{2} \). Its reciprocal would be \( \frac{2}{1} \) or just 2.
This concept arises naturally when handling expressions with negative exponents.
A negative exponent prompts you to take the reciprocal of the base.
This can be expressed generally as follows: if you have a number \( a^{-n} \), its equivalent will be \( \frac{1}{a^{n}} \).
Taking the reciprocal shifts the exponent from negative to positive, making it easier to compute.
Simplifying Expressions
Simplifying expressions can often make complex mathematical problems more manageable.
When you simplify an expression, such as \( \frac{2}{1} \), you reduce it to its most basic form.
In this case, \( \frac{2}{1} \) simplifies to simply 2, because any number divided by 1 is itself.
Sometimes simplification may involve different steps depending on the mathematical operation involved.
With negative exponents, simplification often follows taking the reciprocal, reducing the fraction to its simplest form.
This process not only makes the expression cleaner but also easier to interpret and calculate in further mathematical operations.
When you simplify an expression, such as \( \frac{2}{1} \), you reduce it to its most basic form.
In this case, \( \frac{2}{1} \) simplifies to simply 2, because any number divided by 1 is itself.
Sometimes simplification may involve different steps depending on the mathematical operation involved.
With negative exponents, simplification often follows taking the reciprocal, reducing the fraction to its simplest form.
This process not only makes the expression cleaner but also easier to interpret and calculate in further mathematical operations.
Rational Numbers
Rational numbers are a vital concept in mathematics, defining numbers that can be expressed as the quotient of two integers.
Typically, they take the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
Every integer is a rational number because it can be written as itself divided by 1.
For example, the integer 2 can be expressed as the rational number \( \frac{2}{1} \).
Understanding rational numbers is crucial when simplifying expressions, especially those with negative exponents.
The expression \( \left( \frac{1}{2} \right)^{-1} \) is a perfect illustration of how working with reciprocals and simplification are essential.
By identifying \( \left( \frac{1}{2} \right)^{-1} \) as a rational number and simplifying it properly, we reveal its true identity as the number 2.
Typically, they take the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).
Every integer is a rational number because it can be written as itself divided by 1.
For example, the integer 2 can be expressed as the rational number \( \frac{2}{1} \).
Understanding rational numbers is crucial when simplifying expressions, especially those with negative exponents.
The expression \( \left( \frac{1}{2} \right)^{-1} \) is a perfect illustration of how working with reciprocals and simplification are essential.
By identifying \( \left( \frac{1}{2} \right)^{-1} \) as a rational number and simplifying it properly, we reveal its true identity as the number 2.
Other exercises in this chapter
Problem 5
Simplify each expression. In each exercise, all variables are positive. \(x^{6} \div x^{2}\)
View solution Problem 6
In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x e^{3}=e^{4} $$
View solution Problem 6
In \(3-37,\) express each power as a rational number in simplest form. $$ 8^{\frac{1}{3}} $$
View solution Problem 6
Write each number as a power. 49
View solution