/*! 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 5 Solve \(2^{x^{2}}=8\) for \(x\).... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 5

Solve \(2^{x^{2}}=8\) for \(x\).

Short Answer

Expert verified
The solutions are \(x = \sqrt{3}\) and \(x = -\sqrt{3}\).

Step by step solution

01

Rewrite the Equation with the Same Base

The equation is given as \(2^{x^{2}} = 8\). Notice that 8 is a power of 2, specifically \(8 = 2^3\). Rewrite the equation using this same base: \(2^{x^{2}} = 2^{3}\).
02

Equate the Exponents

Since the bases are the same, we can equate the exponents. This gives us the equation \(x^{2} = 3\).
03

Solve for x

Solve the equation \(x^{2} = 3\) by taking the square root of both sides. Remember to consider both the positive and negative roots: \(x = \sqrt{3}\) or \(x = -\sqrt{3}\).

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Key Concepts

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

Solving Exponential Equations
An exponential equation is one where the variable is found in the exponent. To solve these types of equations, the key is often to express both sides with the same base. This makes it easier to compare and find the value of the unknown variable. For example, in the equation \(2^{x^2} = 8\), we can observe that 8 can be written as a power of 2; that is \(8 = 2^3\). By rewriting the equation as \(2^{x^2} = 2^3\), we set up the equation to equate the exponents directly because the bases are the same.
  • Rewrite each term to possess the same base, if possible.
  • Once the bases are equal, equate the exponents.
  • Solve the resulting equation for the variable.
Pay attention to the possibility that the variable could have multiple values鈥攖his is crucial in correctly solving exponential equations.
Exponent Laws
Exponent laws are fundamental rules that simplify the manipulation of expressions involving powers. These laws are particularly useful when working with exponential equations where simplification is necessary.
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m \times n}\)
  • Power of a Product: \((ab)^m = a^m \times b^m\)
  • Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)
Applying these laws helps in rewriting terms to the necessary form. For example, recognizing that \(8 = 2^3\) allows us to set equations to the same base, thus simplifying further algebraic steps. Knowing and using exponent laws efficiently makes solving complex problems much easier.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. When solving equations like \(x^2 = 3\), finding the square root is a common step to isolate \(x\). Remember to consider both the positive and negative roots since both \(x = \sqrt{3}\) and \(x = -\sqrt{3}\) satisfy the equation.
  • Principal Square Root: Typically indicated by \(\sqrt{}\), the non-negative value.
  • When dealing with equations, practice taking the square root of both sides to solve for the unknown.
  • Always remember that \(x^2 = a\) has solutions \(x = \sqrt{a}\) and \(x = -\sqrt{a}\).
Understanding square roots is crucial in many areas of algebra, as they often represent critical solutions to polynomial equations. Practicing how to find square roots manually will sharpen problem-solving skills across a variety of mathematical contexts.

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Most popular questions from this chapter

Suppose \(f(x)=3 x-9\) and \(g(x)=\sqrt{x}\). What is the domain of the composition \((g \circ f)(x) ?\)

Sketch the graph of \(f(x)=\left\\{\begin{array}{cc}-x & \text { if } x \leq 0 \\\ \tan ^{-1} x & \text { if } x>0\end{array}\right.\)

Find the domain of each of the following functions:(a) \(y=x^{2}+1\) (b) \(y=f(x)=\sqrt{2 x-3}\) (c) \(y=f(x)=1 /(x+1)\) (d) \(y=f(x)=1 /\left(x^{2}-1\right)\) (e) \(y=f(x)=\sqrt{-1 / x}\) (f) \(y=f(x)=\sqrt[3]{x}\) (g) \(y=f(x)=\sqrt{r^{2}-(x-h)^{2}},\) where \(r\) and hare positive constants. (h) \(y=f(x)=\sqrt[4]{x}\) (i) \(y=\sqrt{1-x^{2}}\) (j) \(y=f(x)=\sqrt{1-(1 / x)}\) (k) \(y=f(x)=1 / \sqrt{1-(3 x)^{2}}\) (l) \(y=f(x)=\sqrt{x}+1 /(x-1)\) (m) \(y=f(x)=1 /(\sqrt{x}-1)\)

Expand \(\log _{2} \frac{x^{3}}{3 x-5+(7 / x)}\)

Starting with the graph of \(y=\sqrt{x},\) the graph of \(y=1 / x,\) and the graph of \(y=\sqrt{1-x^{2}}\) (the upper unit semicircle), sketch the graph of each of the following functions: (a) \(f(x)=\sqrt{x-2}\) (b) \(f(x)=-1-1 /(x+2)\) (c) \(f(x)=4+\sqrt{x+2}\) (d) \(y=f(x)=x /(1-x)\) (e) \(y=f(x)=-\sqrt{-x}\) (f) \(f(x)=2+\sqrt{1-(x-1)^{2}}\) (g) \(f(x)=-4+\sqrt{-(x-2)}\) (h) \(f(x)=2 \sqrt{1-(x / 3)^{2}}\) (i) \(f(x)=1 /(x+1)\) (j) \(f(x)=4+2 \sqrt{1-(x-5)^{2} / 9}\) (k) \(f(x)=1+1 /(x-1)\) (l) \(f(x)=\sqrt{100-25(x-1)^{2}}+2\)
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