Exponents Reference Sheet

Today I’m posting the first installment of a new occasional series for this blog. In each Reference Sheet, I’ll cover, as succinctly as possible, every rule you absolutely must know to solve problems in a single area found on standardized tests.

Learning these rules isn’t a substitute for developing higher-order problem-solving and strategic thinking skills; rather, it’s a necessary precondition and foundation for all of that strategizing to take place. This Reference Sheet lists the minimum requirements to get your foot in the door. It’s the price of admission.

If you’re taking any test involving exponents, here’s what you need to know:

 


 

Exponent Basics:

For any real numbers x and A:

x^A means x*x*...*x    (\leftarrowA copies of x, multiplied together)

 \begin{tabular*}{0.6\textwidth}{@{\extracolsep{\fill}} l r } 0^A=0 & x^{-A}=\frac{1}{x^A}\\ 1^A=1 & x^{\frac{1}{A}}=\sqrt[A]{x}\\ x^0=1 & \sqrt{x}=\sqrt[2]{x}=x^{\frac{1}{2}}\\ x^1=x & \\ \end{tabular}

Note: Roots are just fractional exponents. There are no root rules; there are only exponent rules, and all exponent rules apply as well to roots.

Note: 0^0 is indeterminate/undefined and will not be tested.

Three Big Exponent Rules:

I. x^a*x^b=x^{a+b}

Bonus: \frac{x^a}{x^b}=x^{a-b}

II. (x^a)^b=(x^b)^a=x^{ab}

III. (xy)^a=x^a*y^a

Bonus: (\frac{x}{y})^a=\frac{x^a}{y^a}

Four (Plus One) Steps to Solving an Exponents Problem:

1. Prime factor your bases.

2. If bases are added or subtracted, factor/”count apples” to collect added bases.

3. Collect like bases using exponent rules.

4. Set exponents of like bases equal.

5. Solve the resulting equation(s).

 


 

Example Problems:

Example 1. For integers x and y, \frac{1}{\sqrt[x]{12}}*15^{2y}=2160. What is the value of x+y?

 

Solution

Step 1: Prime factor your bases.

  • \frac{1}{\sqrt[x]{2^2*3}}*(3*5)^{2y}=2^4*3^3*5^1

Step 3: Collect like bases using exponent rules.

  • (2^2*3)^{-\frac{1}{x}}*3^{2y}*5^{2y}=2^4*3^3*5^1 (using Rule III)
  • 2^{-\frac{2}{x}}*3^{-\frac{1}{x}}*3^{2y}*5^{2y}=2^4*3^3*5^1 (using Rule II and Rule III)
  • 2^{-\frac{2}{x}}*3^{2y-\frac{1}{x}}*5^{2y}=2^4*3^3*5^1 (using Rule I)

Step 4: Set exponents of like bases equal.

  • -\frac{2}{x}=4; 2y-\frac{1}{x}=3; 2y=1

Step 5: Solve the resulting equation(s).

  • y=\frac{1}{2}
  • -\frac{2}{x}=4; \frac{1}{x}=-2
  • x=-\frac{1}{2}
  • x+y=-\frac{1}{2}+\frac{1}{2}=0

 

Example 2. For what integer x does -2^{x-2}+2^{x-1}-2^x+2^{x+1}=160?

 

Solution

Step 2: If bases are added or subtracted, factor/”count apples” to collect added bases.

  • -(2^{-2}*2^x)+(2^{-1}*2^x)-(2^x)+(2^1*2^x)=160 (using Rule I in reverse)
  • -\frac{1}{4}*2^x+\frac{1}{2}*2^x-1*2^x+2*2^x=160
  • (-\frac{1}{4}+\frac{1}{2}-1+2)*2^x=160
  • (-\frac{1}{4}+\frac{2}{4}-\frac{4}{4}+\frac{8}{4})*2^x=160
  • \frac{5}{4}*2^x=160

Step 1: Prime factor your bases.

  • 5*2^{-2}*2^x=5*2^5

Step 3: Collect like bases using exponent rules.

  • 5*2^{x-2}=5*2^5 (using Rule I)

Step 4: Set exponents of like bases equal.

  • x-2=5

Step 5: Solve the resulting equation(s).

  • x=7

 

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