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Chapter 0: Problem 98

Simplify the expressions. $$ 2^{1 / a} / 2^{2 / a} $$

Short Answer

Expert verified
The simplified expression is \(2^{\frac{-1}{a}}\).

Step by step solution

01

Identify the rule for dividing exponents

When we have a division of two numbers with the same base and with exponents, we can use the rule for dividing exponents: \(b^{m} / b^{n} = b^{m - n}\) In this case, our base is 2 and the exponents are \(\frac{1}{a}\) and \(\frac{2}{a}\).
02

Apply the rule to the given expression

Now that we know the rule, we can apply it to the given expression: \[ 2^{\frac{1}{a}} / 2^{\frac{2}{a}} = 2^{\frac{1}{a} - \frac{2}{a}} \]
03

Simplify the exponent expression

The next step is to simplify the exponent term in the expression. We can do this by finding a common denominator for the two fractions and then subtracting them: \[ \frac{1}{a} - \frac{2}{a} = \frac{1 - 2}{a} = \frac{-1}{a} \]
04

Substitute the simplified exponent back into the expression

Now that we have simplified the exponent, we can substitute it back into the original expression: \[ 2^{\frac{1}{a} - \frac{2}{a}} = 2^{\frac{-1}{a}} \] So, the simplified expression is: \[ 2^{\frac{-1}{a}} \]

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Most popular questions from this chapter

By any method, determine all possible real solutions of each equation Check your answers by substitution. \(x+4=\frac{1}{x-2}\)

Find all possible real solutions of each equation \(x^{3}-x^{2}-3 x+3=0\)

Convert each expression in Exercises into its technology formula equivalent as in the table in the text.\(2.1 x^{-3}-x^{-1}+\frac{x^{2}-3}{2}\)

Solve the equation for \(x\) (mentally, if possible). \(a x+b=c \quad(a \neq 0)\)

By any method, determine all possible real solutions of each equation Check your answers by substitution. \(x^{2}+x+1=0\)
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