Exponential Equations - GMAT Math Study Guide

Definitions

• Base - the number that is multiplied by itself a certain quantity of times.
  For example, in the expression 2³, the number 2 is the base.
• Exponent - the number of times a quantity is multiplied by itself.
  For example, in the expression 2³, the number 3 is the exponent.
• Radical - the sign used to denote the square or n^th root of a number.
  For example, the value of "radical 4" is 2 and the value of "radical 9" is 3.
• Exponential Expression - an expression or term with a power or exponent that is not one.
  For example, x² is an exponential expression while x is not an exponential expression. Similarly, x^1/2 (called the square root of x) is an exponential expression while 2x is not an exponential expression.
• Exponential Equation - an equation with a term that has an exponent greater than one.
  For example, x^3/2 + 2x + 1 is an exponential expression while 2x + 3 is not an exponential expression. Similarly, x³ = 27 is an exponential equation while x + 2 = 29 is not an exponential equation.

Exponents signify repeated self-multiplication. E.g.,: 2³ = 2*2*2

Exponents are a shorthand way of representing repeated multiplication. Consider the following examples, which are all exponential equations because a term is multiplied by itself multiple times:

Laws of Exponents

There are many laws of exponents that should be memorized and practiced in order to be thoroughly understood. The following exponent laws are detailed more thoroughly with examples on the exponential powers page and the radicals & roots page.

Solving Exponential Equations

Exponential Equations & the Number of Solutions

One property of exponential equations that is initially confusing to some students is determining how many solutions an equation will have.

1. Exponential equations with one term and an even power will have up to 2 solutions.
   
   x⁴=-2 has no solutions
   x⁴=16 has solutions x=-2 and x=2
2. Exponential equations with one term and an odd power will have exactly one solution.
   
   x³=27 has one solution x=3. Notice x=-3 is not a solution as (-3)*(-3)*(-3) = -27
3. Exponential equations with multiple terms and both even and odd exponents can have many solutions.
   
   x³ - 7x + 6 = 0 has three solutions: x=1, 2, -3

An even exponent hides the sign of its roots (e.g., x²=4; x = 2 and x = -2).

One of the more commonly tested properties of exponents and exponential equations is that an even exponent hides the sign of its roots. Consider the following example:

Techniques for Solving Exponential Equations

As noted above, an exponential equation has one or more terms with a base that is raised to a power that is not 1. While there is no formula for solving an exponential equation, the following examples provide some insight into common techniques used in finding the unknown value in an exponential equation.

Technique 1: Isolate and Raise to the Inverse Exponent

Arrange the term with an exponent on one side of the equation and the other terms on the other side of the equation. Raise both sides of the equation to the inverse exponent.

Work to isolate the x⁴ term by subtracting 6 from both sides and then dividing both sides by 3.

In order to isolate x, since x is raised to the 4/1 power, raise both sides to the inverse power (i.e., 1/4).

Technique 2: Solve Through Factoring

Isolating an exponent often makes solving an equation easier.

For a more detailed explanation of this technique, please visit the factoring study guide and the quadratic equations study guide. Arrange all similar terms on one side of the equal sign and then factor.

Divide each term by 2, which is a common factor, and then subtract the number on the right side of the equation.

Using factoring rules, simplify and solve the exponential equation.

Multiplying Exponents

The formulas above can be used to multiply together exponents in order to solve exponential equations. The most important formulas to use are the following:

In order to better understand how these formulas could be used to multiply exponents, consider the following example:

Dividing Exponents

The aforementioned formulas are helpful in dividing exponents, especially these two exponent formulas.

In order to better understand the division of exponents, consider the following example, which is solved in two ways to provide a more thorough understanding of how to divide exponents.

Method 1

Method 2

Simplifying Exponents

The above examples provide some insight into the process of simplifying exponents. While there are no hard and fast rules for this process, asking the following questions often provides clues about how to best simplify an expression.

1. Are there terms that can be written with a common base or exponent?
2. Are there terms that can be reduced or canceled?
3. Can terms be factored so as to yield a common term or solution?

Types of GMAT Problems

1. Solving an Exponential Equation

   Basic exponential equations can be solved by isolating the term with the exponent. For an exponent m/n, take the n/m exponent of both sides, simplify, and solve for x.

   Solve for x, x⁵-32 = 0

   A)	3
   B)	2
   C)	7
   D)	5
   E)	4

   Correct Answer: B

   1. Isolate the x⁵ term.
      x⁵ = 32 (add 32 to both sides)
      
   2. Exponentiate both sides by (1/5).
      (x⁵)^(1/5) = (x^5/5) = 32^(1/5)
   3. Since 32 = 2⁵, you can simplify 32^(1/5)
      32^(1/5) = (2⁵)^(1/5)
      
   4. Returning to the originally simplified expression:
      (x^5/5) = 32^(1/5)
      x¹ = 2^(5/5) = 2¹
      x=2
2. Same Base Problems

   When the unknown is in the exponent, it may require rewriting the equation so each side has the same base. If, on each side of the equation, there is only the same constant base with different exponents, a new equation can be written equating the exponents from the two sides.

   Solve for x, (8^2x)² = 64^x-2

   A)	-3
   B)	5
   C)	-1
   D)	4
   E)	-2

   Correct Answer: E

   1. You can solve this problem by using a common base of either 2 or 8
   2. Solve Using a Base of 2
      1. Rewrite both sides to have the same base since both sides are multiples of 2.
         8 = 2³
         64 = 2⁶
         
         (8^2x)² = 64^x-2
         ((2³)^2x)² = (2⁶)^x-2
      2. Simplify by using the laws of exponents.
         (2³)^2x*2 = 2^6*(x-2)
         2^3*4x = 2^6x-12
         2^12x = 2^6x-12
         
      3. Take out the bases and equate the exponents. Solve for x:
         12x = 6x-12
         6x = -12 (subtract 6x from both sides)
         x = -2 (divide both sides by 6)
   3. Solve Using a Base of 8
      1. Rewrite both sides to have the same base since both sides are multiples of 8.
         (8^2x)² = ((8¹)^2x)² = 8^4x
         64^x-2 = (8²)^x-2 = 8^{2x - 4}
         
         8^4x = 8^{2x - 4}
      2. Take out the bases and equate the exponents. Solve for x:
         4x = 2x - 4
         2x = -4
         x = -2
3. Eliminating Roots

   If a problem contains a square root, it is easiest to isolate the radical term and then square both sides. Also, for equations with cube roots, the same can be done, except cube both sides rather than square them.

   

   A)	x=6 or x=-6
   B)	x=6
   C)	x=5 or x=-5
   D)	x=4
   E)	x=4 or x=-4

   Correct Answer: A

   1. Isolate the radical.
      (add 7 to both sides)
   2. Square both sides.
      x²+13 = 49
   3. Solve for x.
      x² = 36 (subtract 13 from both sides)
      x=6 or x=-6 (take the square root of both sides remembering to include both the positive and negative solutions)

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