If $a=1+2^n$ And $b=1+2^{-n}$, Show That $ B = A A − 1 B=\frac{a}{a-1} B = A − 1 A ​ [/tex].

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If $a=1+2^n$ and $b=1+2^{-n}$, show that $b=aa−1b=\frac{a}{a-1}

In this article, we will explore a mathematical problem involving two variables, a and b, defined in terms of powers of 2. We will show that the variable b can be expressed as a fraction of a, specifically $b=aa−1b=\frac{a}{a-1}$. This problem requires us to manipulate the given expressions for a and b, and apply algebraic techniques to derive the desired result.

We are given two expressions:

a=1+2na=1+2^n

b=1+2−nb=1+2^{-n}

where n is a positive integer.

To show that $b=aa−1b=\frac{a}{a-1}$, we will start by manipulating the expression for b.

Step 1: Multiply b by (a-1)

We can multiply the expression for b by (a-1) to get:

b(a−1)=(1+2−n)(a−1)b(a-1) = (1+2^{-n})(a-1)

Step 2: Expand the right-hand side

Expanding the right-hand side of the equation, we get:

b(a−1)=1(a−1)+2−n(a−1)b(a-1) = 1(a-1) + 2^{-n}(a-1)

Step 3: Simplify the right-hand side

Simplifying the right-hand side of the equation, we get:

b(a−1)=a−1+2−na−2−nb(a-1) = a - 1 + 2^{-n}a - 2^{-n}

Step 4: Factor out a

Factoring out a from the first two terms on the right-hand side, we get:

b(a−1)=a(1+2−n)−1−2−nb(a-1) = a(1 + 2^{-n}) - 1 - 2^{-n}

Step 5: Simplify the right-hand side

Simplifying the right-hand side of the equation, we get:

b(a−1)=a(1+2−n)−(1+2−n)b(a-1) = a(1 + 2^{-n}) - (1 + 2^{-n})

Step 6: Cancel out the common term

Canceling out the common term (1 + 2^{-n}) on the right-hand side, we get:

b(a−1)=a−1b(a-1) = a - 1

Step 7: Divide both sides by (a-1)

Dividing both sides of the equation by (a-1), we get:

b=a−1a−1b = \frac{a - 1}{a - 1}

Step 8: Simplify the right-hand side

Simplifying the right-hand side of the equation, we get:

b=aa−1b = \frac{a}{a - 1}

In this article, we have shown that the variable b can be expressed as a fraction of a, specifically $b=aa−1b=\frac{a}{a-1}$. We started by manipulating the expression for b, and applied algebraic techniques to derive the desired result. This problem requires a good understanding of algebraic manipulation and fraction simplification.

  • The variable b can be expressed as a fraction of a, specifically $b=aa−1b=\frac{a}{a-1}$.
  • The expression for b can be manipulated using algebraic techniques.
  • Fraction simplification is an important skill in algebra.

For further reading on algebraic manipulation and fraction simplification, we recommend the following resources:

  • Algebra
  • Fraction
    Q&A: If $a=1+2^n$ and $b=1+2^{-n}$, show that $b=aa−1b=\frac{a}{a-1}$

In our previous article, we showed that the variable b can be expressed as a fraction of a, specifically $b=aa−1b=\frac{a}{a-1}$. In this article, we will answer some common questions related to this problem.

Q: What is the significance of the expressions $a=1+2^n$ and $b=1+2^{-n}$?

A: The expressions $a=1+2^n$ and $b=1+2^{-n}$ are used to define the variables a and b in terms of powers of 2. The variable n is a positive integer.

Q: How do we manipulate the expression for b to show that $b=aa−1b=\frac{a}{a-1}$?

A: To show that $b=aa−1b=\frac{a}{a-1}$, we start by multiplying the expression for b by (a-1). We then expand and simplify the right-hand side of the equation to get the desired result.

Q: What is the key step in the derivation of $b=aa−1b=\frac{a}{a-1}$?

A: The key step in the derivation of $b=aa−1b=\frac{a}{a-1}$ is the cancellation of the common term (1 + 2^{-n}) on the right-hand side of the equation.

Q: Can we use this result to find the value of b in terms of a?

A: Yes, we can use this result to find the value of b in terms of a. By substituting the expression for a into the equation $b=aa−1b=\frac{a}{a-1}$, we can find the value of b in terms of a.

Q: What are some common applications of this result?

A: This result has many common applications in mathematics and computer science. For example, it can be used to simplify expressions involving powers of 2, and to find the value of b in terms of a.

Q: How can we generalize this result to other expressions involving powers of 2?

A: We can generalize this result to other expressions involving powers of 2 by using similar algebraic techniques. For example, we can use the same method to show that $b=aa−1b=\frac{a}{a-1}$ for other expressions involving powers of 2.

Q: What are some common mistakes to avoid when working with expressions involving powers of 2?

A: Some common mistakes to avoid when working with expressions involving powers of 2 include:

  • Not simplifying the expression enough
  • Not canceling out common terms
  • Not using the correct algebraic techniques

In this article, we have answered some common questions related to the problem of showing that $b=aa−1b=\frac{a}{a-1}$. We have also provided some tips and tricks for working with expressions involving powers of 2.

  • The variable b can be expressed as a fraction of a, specifically $b=aa−1b=\frac{a}{a-1}$.
  • The expression for b can be manipulated using algebraic techniques.
  • Fraction simplification is an important skill in algebra.
  • This result has many common applications in mathematics and computer science.

For further reading on algebraic manipulation and fraction simplification, we recommend the following resources: