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Chapter 2: Problem 32

\(n^{\frac{5}{2}}=32, \text { what is the value of } n ?\)

Short Answer

Expert verified
The value of \(n\) is 4.

Step by step solution

01

- Understand the Given Equation

The given equation is \(n^{\frac{5}{2}} = 32\). This means that the power of \(n\) to the fraction \(\frac{5}{2}\) equals 32.
02

- Isolate the Variable

To find \(n\), we need to isolate it. To do this, we'll raise both sides of the equation to the power of \(\frac{2}{5}\) to counter the \(\frac{5}{2}\) exponent.
03

- Apply the Exponent

Raise both sides to the power of \(\frac{2}{5}\): \[ (n^{\frac{5}{2}})^{\frac{2}{5}} = 32^{\frac{2}{5}} \] This simplifies to: \(n = 32^{\frac{2}{5}}\).
04

- Simplify the Term

Simplify the right side of the equation: \[ 32 = 2^5 \] So, \[ 32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^{5 \cdot \frac{2}{5}} = 2^2 = 4 \]
05

- Solution

Hence, the value of \(n\) is 4.

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Key Concepts

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

Understanding Exponentiation
Exponentiation is a mathematical operation where a number, called the base, is raised to a power, called the exponent. For example, in the expression \[ a^b \], 'a' is the base and 'b' is the exponent. This means that 'a' is multiplied by itself 'b' times. In our exercise, we start with \[ n^{\frac{5}{2}} \]. In fractional exponents, such as \[ n^{\frac{5}{2}} \], the numerator (5) indicates the power, and the denominator (2) indicates the root. In simple terms, \[ n^{\frac{5}{2}} \] is equivalent to \[ (n^5)^{\frac{1}{2}} \], or the square root of \( n^5 \). Using this understanding can make solving such problems easier.
Isolating Variables
Isolating the variable is a crucial step in solving algebraic equations. It involves performing operations that will leave the variable alone on one side of the equation. In the given problem, we have \[ n^{\frac{5}{2}} = 32 \]. To isolate 'n', we need to counteract the \[ \frac{5}{2} \] exponent by using its reciprocal, which is \[ \frac{2}{5} \].
  • Raise both sides of the equation to the power of \[ \frac{2}{5} \].
  • This action will cancel out the original exponent on 'n', simplifying our equation.
Now, when we apply this to our problem, we get \[ (n^{\frac{5}{2}})^{\frac{2}{5}} = 32^{\frac{2}{5}} \]. Thus, \[ n = 32^{\frac{2}{5}} \].
Simplifying Expressions
Simplifying expressions involves breaking down complex equations into simpler forms. This often requires a good understanding of exponent rules. In our problem, after isolating 'n', we get \[ n = 32^{\frac{2}{5}} \]. We can simplify \[ 32^{\frac{2}{5}} \] by recognizing that 32 is a power of 2: \[ 32 = 2^5 \].
  • Replace 32 with \[ 2^5 \]: \[ 32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} \].
  • Use the exponent multiplication rule \[ (a^m)^n = a^{m \cdot n} \]: \[ (2^5)^{\frac{2}{5}} = 2^{5 \cdot \frac{2}{5}} \].
  • Simplify the expression \[ 2^{5 \cdot \frac{2}{5}} = 2^2 = 4 \].
Thus, \[ n = 4 \].
Solving Algebraic Equations
Solving algebraic equations involves finding the value(s) of the variables that satisfy the equation. In this exercise, our goal is to find the value of 'n' that satisfies \[ n^{\frac{5}{2}} = 32 \]. Here's a step-by-step approach:
  • Identify the equation and understand the relationship between the base and the exponent.
  • Isolate the variable by applying reciprocal exponentiation to both sides of the equation.
  • Simplify the resulting expression, using exponent rules where necessary.
By following these steps, we found that \[ n = 4 \]. This systematic approach helps in tackling similar algebraic problems efficiently.

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