/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 99 Solve. $$m^{2 / 3}-2 m^{1 / 3}... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 99

Solve. $$m^{2 / 3}-2 m^{1 / 3}-8=0$$

Short Answer

Expert verified
The solutions are \( m = 64 \) and \( m = -8 \).

Step by step solution

01

Substitute a New Variable

Let us introduce a new variable to simplify the equation. Let\( n = m^{1/3} \). Then, the given equation\( m^{2/3} - 2m^{1/3} - 8 = 0 \) can be rewritten as \( n^2 - 2n - 8 = 0 \).
02

Solve the Quadratic Equation

Now, solve the quadratic equation \( n^2 - 2n - 8 = 0 \). This can be factored as \( (n - 4)(n + 2) = 0 \). This gives the solutions \( n = 4 \) and \( n = -2 \).
03

Substitute Back the Original Variable

Revert to the original variable by substituting \( n \) back with \( m^{1/3} \). Therefore, we get two equations: \( m^{1/3} = 4 \) and \( m^{1/3} = -2 \).
04

Solve for m

Solve for \( m \) in each equation. From \( m^{1/3} = 4 \), we get \( m = 4^3 = 64 \). From \( m^{1/3} = -2 \), we get \( m = (-2)^3 = -8 \).

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Key Concepts

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

substitution method
The substitution method simplifies complex equations by introducing a new variable to replace an expression. This method can transform an equation into a simpler form, making it easier to solve.
  • Identify a complex expression in the equation.
  • Introduce a new variable to replace this expression.
  • Rewrite the original equation using the new variable.
In this exercise, the substitution was done by setting \( n = m^{1/3} \). This effectively changed the original equation \( m^{2 / 3} - 2 m^{1 / 3} - 8=0 \) to a more recognizable quadratic form \( n^2 - 2n - 8 = 0 \). The purpose is to reduce complexity and work with simpler equations.
quadratic equation
Quadratic equations are polynomial equations of degree 2, generally in the form \( ax^2 + bx + c = 0 \). Here, our equation \((n-4)(n+2)=0\) can be factored into two binomials:
  • If \( ax^2 + bx + c = 0 \), it can be written as \((px + q)(rx + s) = 0 \).
  • Solve for \( n\): \( n^2 - 2n - 8 = 0 \) factors to \( (n-4)(n+2) = 0 \).
From the factored form \( (n-4)(n+2) = 0 \), we find the solutions by setting each factor to zero \( n-4=0 \) and \( n+2=0 \), yielding the solutions \( n=4 \) and \( n=-2 \).
factoring
Factoring is the process of breaking down a polynomial into simpler components, or 'factors,' that, when multiplied together, yield the original polynomial. This is an essential skill for solving polynomial equations.
  • Identify common factors in terms.
  • Rewrite the equation as a product of these factors.
  • Solve the simpler equations.
In our exercise, the quadratic equation \( n^2 - 2n - 8 = 0 \) was factored into \( (n - 4)(n + 2) = 0 \). By solving each factor separately, we obtained the solutions \( n = 4 \) and \( n = -2 \).
Factoring simplifies solving higher-degree polynomials by breaking them down into simpler, easily solvable terms.

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