Question

Explain the mistake that is made. Evaluate the function for the given $x: f(x)=4^{x}$ for $x=\frac{3}{2}$. $\begin{aligned} \text { Solution: } & f\left(\frac{3}{2}\right)=4^{3 / 2} \\ &=\frac{4^{3}}{4^{2}}=\frac{64}{16}=4 \end{aligned}$ The correct value is $8 .$ What mistake was made?

Name: Problem 70, Chapter 3: Precalculus
Uploaded: 2020-08-10T17:39:59-08:00
Duration: 32 s
Channel: Katelyn Chen
Description: Explain the mistake that is made. Evaluate the function for the given x: f(x)=4^x for x=(3)/(2). Solution: f((3)/(2))=4^3 / 2 =(4^3)/(4^2)=(64)/(16)=4 The correct value is 8 . What mistake was made?

   Explain the mistake that is made.
Evaluate the function for the given $x: f(x)=4^{x}$ for $x=\frac{3}{2}$.
$\begin{aligned} \text { Solution: } & f\left(\frac{3}{2}\right)=4^{3 / 2} \\ &=\frac{4^{3}}{4^{2}}=\frac{64}{16}=4 \end{aligned}$
The correct value is $8 .$ What mistake was made?

Precalculus

Cynthia Young 2nd Edition

Chapter 3, Problem 70

Instant Answer

Solved by Expert Katelyn Chen

08/10/2020

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Step 1: We are given the function $f(x)=4^{x}$ and we need to evaluate it for $x=\frac{3}{2}$. Show more…

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Explain the mistake that is made. Evaluate the function for the given $x: f(x)=4^{x}$ for $x=\frac{3}{2}$. $\begin{aligned} \text { Solution: } & f\left(\frac{3}{2}\right)=4^{3 / 2} \\ &=\frac{4^{3}}{4^{2}}=\frac{64}{16}=4 \end{aligned}$ The correct value is $8 .$ What mistake was made?

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Key Concepts

Fractional Exponents

Fractional exponents represent roots as well as powers. For example, an expression of the form a^(m/n) is interpreted as the nth root of a raised to the mth power, or equivalently, (nth root of a)^m. This definition aligns with the idea that exponents must be handled as operations involving both exponentiation and extraction of roots, rather than being split into separate numerator and denominator exponent operations.

Properties of Exponents

The laws of exponents, such as a^(m) / a^(n) = a^(m - n), are applicable only when the exponents are whole numbers or when the expressions involved allow such cancellation. In the given solution, the mistake was to incorrectly apply this property by writing 4^(3/2) as 4^3 divided by 4^2, which falsely assumes that the numerator and denominator of the exponent can be separated into 4^(3) and 4^(2). This misuse leads to an incorrect result because the rule does not apply to fractional exponents in that manner.

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