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Chapter 0: Problem 25

Compute the numbers. $$4^{-1 / 2}$$

Short Answer

Expert verified
\( \frac{1}{2} \)

Step by step solution

01

Understanding the Negative Exponent

A negative exponent indicates the reciprocal of the base raised to the positive of the exponent. So, rewrite the expression to get rid of the negative sign: \[ 4^{-1/2} = \frac{1}{4^{1/2}} \]
02

Understanding the Fractional Exponent

The fractional exponent \( \frac{1}{2} \) represents a square root. Hence, rewrite the expression using the square root notation:\[ \frac{1}{4^{1/2}} = \frac{1}{\sqrt{4}} \]
03

Calculating the Square Root

Calculate the square root of 4 which is 2:\[ \frac{1}{\sqrt{4}} = \frac{1}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponents
Negative exponents can seem tricky at first, but they're not too hard to grasp! A negative exponent means that instead of multiplying, you should divide. In other words, you take the reciprocal. For example, if you have the expression \(a^{-n}\), it is equal to \(\frac{1}{a^n}\).

That means you flip the base to the bottom of a fraction and turn the exponent positive. Let's see this in action with our example: \[4^{-1/2} = \frac{1}{4^{1/2}} = \frac{1}{\sqrt{4}}\]
Notice how we moved the base 4 to the denominator and changed the negative exponent to positive?
Square Roots
Square roots indicate a value that, when multiplied by itself, gives the original number. You use the symbol \(\sqrt{}\) for square roots. For example, \(\sqrt{9} = 3\), because 3 multiplied by 3 equals 9.

In our problem, we saw a fractional exponent \(\frac{1}{2}\), which means the square root. When we process \(4^{1/2}\), it's the same as finding \(\sqrt{4}\). The result of \(\sqrt{4} \)is 2.
Reciprocals
Reciprocals are numbers that, when multiplied by the original number, result in 1. Simply put, the reciprocal of a number is \(\frac{1}{number}\). For example, the reciprocal of 5 is \(\frac{1}{5}\), and the reciprocal of \(\frac{3}{4}\)is \(\frac{4}{3}\).

In our problem, after applying the negative exponent, we found \(\frac{1}{4^{1/2}}\). This resulted in \(\frac{1}{2}\). So every time you see a negative exponent, remember that the reciprocal turns it positive and places it in the denominator.

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Most popular questions from this chapter

Convert the numbers from graphing calculator form to standard form (that is, without E). $$8.23 E-6$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{x^{3}}{y^{-2}}$$

Let \(f(x)=x^{2}+3 x+1\) and \(\operatorname{let} g(x)=x^{2}-3 x-1 .\) Graph the two functions \(f(g(x))\) and \(g(f(x))\) together in the window [-4,4] by [-10,10] and determine if they are the same function.

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\frac{x^{4} \cdot y^{5}}{x y^{2}}$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\left(x^{1 / 3}\right)^{6}$$
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