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Chapter 11: Problem 66

Solve the given problems.Solve for $$x: 3^{-1} x^{2}=-9 x^{-1}$$.

Short Answer

Expert verified
The solution is \(x = -3\).

Step by step solution

01

Simplify the Equation

The given equation is \(3^{-1} x^{2} = -9 x^{-1}\). Start by simplifying \(3^{-1}\), which is equal to \(\frac{1}{3}\). The equation becomes \(\frac{1}{3} x^{2} = -9 x^{-1}\).
02

Remove the Fraction

Multiply both sides of the equation by 3 to eliminate the fraction. This gives: \[x^2 = -27 x^{-1}\].
03

Simplify Further

Multiply both sides of the equation by \(x\) to clear the negative exponent on the right side:\[x^3 = -27\].
04

Solve for x

To isolate \(x\), take the cube root of both sides of the equation:\[x = \sqrt[3]{-27}\].This simplifies to \(x = -3\).

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Key Concepts

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

Exponents
Exponents are a shorthand way of expressing repeated multiplication. When we see something like \(a^n\), this means \(a\) is multiplied by itself \(n\) times. For example, \(3^4\) is \(3 \times 3 \times 3 \times 3 = 81\). Exponents can also be negative or fractional:
  • Negative exponents, such as \(3^{-1}\), indicate the reciprocal of the base raised to the positive exponent, so \(3^{-1} = \frac{1}{3}\).
  • Fractional exponents, like \(a^{\frac{1}{2}}\), represent roots. \(a^{\frac{1}{2}}\) is equivalent to \(\sqrt{a}\).
Using exponents can simplify equations, especially when dealing with large numbers or roots. It's important to become comfortable using them both in their negative and fractional forms as they frequently appear in mathematical equations.
Simplifying Equations
Simplifying equations is a fundamental skill in algebra. It involves reducing an equation to its simplest form. This makes it easier to solve. The process usually involves:
  • Combining like terms.
  • Clearing fractions by multiplying both sides of the equation by a common denominator.
  • Eliminating exponents by their definitions, such as transforming \(3^{-1}\) into \(\frac{1}{3}\).
For instance, in the equation \(3^{-1} x^{2} = -9 x^{-1}\), we start by changing the fraction \(3^{-1}\) to \(\frac{1}{3}\), resulting in \(\frac{1}{3} x^{2} = -9 x^{-1}\). From there, multiplying both sides by \(3\) clears the fraction, resulting in \(x^{2} = -27 x^{-1}\). This makes the equation more straightforward to solve.
Cube Roots
Cube roots help to undo cubing a number or variable. The cube root of a number \(x\), denoted as \(\sqrt[3]{x}\), is a number \(a\) such that \(a^3 = x\). For example, the cube root of \(8\) is \(2\) because \(2^3 = 8\). Here's how cube roots work:
  • Cube roots can be negative. For example, \(\sqrt[3]{-27} = -3\) because \((-3)^3 = -27\).
  • Unlike square roots, cube roots of negative numbers are real and easy to calculate.
In the original problem, after simplifying to \(x^3 = -27\), we solve for \(x\) by taking the cube root of both sides, which gives \(x = \sqrt[3]{-27}\). This simplifies to \(x = -3\). Remember, taking cube roots is an effective way to find unknowns when they are cubed in equations.

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Most popular questions from this chapter

In Exercises \(53-76,\) solve the given problems. Is it true that, if \(x \neq 0,\left[\left(-x^{-2}\right)^{-2}\right]^{-2}=1 / x^{2} ?\)

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$(-7 x)^{0}$$

In Exercises \(53-76,\) solve the given problems. For what integral value(s) of \(n\) is \(n^{\pi}>\pi^{n} ?\)

Graph the given functions. $$f(t)=t^{-4 / 5}$$

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$\left(n^{-2}-2 n^{-1}\right)^{2}$$
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