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Chapter 13: Problem 16

Evaluate the exponential function \(y=4^{x}\) for the given values of \(x\). $$x=5 / 2$$

Short Answer

Expert verified
The value of \( y = 4^{5/2} \) is 1024.

Step by step solution

01

Substitute x into the function

The given exponential function is \( y = 4^x \). We need to evaluate it for \( x = \frac{5}{2} \). Substitute \( x = \frac{5}{2} \) into the function to get \( y = 4^{\frac{5}{2}} \).
02

Simplify the expression using exponent rules

The expression \( 4^{\frac{5}{2}} \) can be simplified by recognizing it as \( (4^2)^{\frac{5}{4}} \). First compute \( 4^2 = 16 \). So, \( 4^{\frac{5}{2}} = \sqrt{16^5} \).
03

Compute powers and roots

Calculate \( 16^5 \). Start by recognizing that \( 16^5 = (2^4)^5 = 2^{20} \). Now, the square root is \( \sqrt{2^{20}} = 2^{10} \).
04

Calculate the final value

Calculate \( 2^{10} = 1024 \). Therefore, the value of \( y \) when \( x = \frac{5}{2} \) is \( 1024 \).

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Key Concepts

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

Understanding Exponents
Exponents are a mathematical way of expressing repeated multiplication. They consist of two parts: the base and the exponent (also known as the power). When we write something like \( 4^x \), the number \( 4 \) is the base, and \( x \) is the exponent. This expression means you multiply the base \( 4 \) by itself \( x \) number of times. In an expression such as \( 4^{\frac{5}{2}} \), the exponent is a fraction, indicating that both powers and roots are involved.

Common rules of exponents include:
  • Product of Powers: \( a^m \cdot a^n = a^{m+n} \).
  • Power of a Power: \( (a^m)^n = a^{m \cdot n} \).
  • Negative Exponents: \( a^{-n} = \frac{1}{a^n} \).
  • Zero Exponent: \( a^0 = 1 \), given \( a eq 0 \).
These rules help simplify expressions and make computations manageable.
Powers and Roots
Powers and roots are interconnected concepts. Powers refer to multiplying a number by itself a certain number of times, while roots find which number, when raised to a certain power, produces the original number. For instance, in the expression \( 4^{\frac{5}{2}} \), the numerator of the exponent's fraction represents the power, and the denominator signifies the root.

For example:
  • To compute \( 4^2 \), multiply 4 by itself to get 16.
  • Similarly, \( \sqrt{16} \), the square root, asks "what number squared gives 16?" The answer is 4.
When dealing with fractional exponents like \( 4^{\frac{5}{2}} \), break it down by computing the root first (\( 4^{1/2} \) for the square root) and then the power (\( (4^{1/2})^5 \)), or vice versa.
Evaluating Functions
Evaluating functions simply means substituting a given value of the independent variable into the function to find the value of the dependent variable. In the original exercise, the function is \( y = 4^x \) and the given \( x \) is \( \frac{5}{2} \). This involves several steps:

  • First, substitute \( x = \frac{5}{2} \) into the expression, getting \( y = 4^{\frac{5}{2}} \).
  • Use exponent rules to simplify the expression, recognizing \( 4^{\frac{5}{2}} \) as \( (4^2)^{\frac{5}{4}} \). Calculate \( 4^2 = 16 \).
  • Evaluate the intermediate step by calculating the root and power: \( \sqrt{16^5} \), or further simplify using powers of 2: \( 2^{10} \).
Complete the operation to find \( 1024 \) as the value of \( y \). Evaluating functions with precision requires understanding both the function rules and mathematical operations involved.

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