Exponential Expressions


You should expect to see questions on the SAT that test your understanding of exponents. In this section we will be reviewing these two topics:

  • Brief Review of Exponents
  • Laws of Exponents
  • Compound Interest

Let’s look at each of these in more detail:

Brief Review of Exponents

When a number is raised to a power, exponents are involved. Here are the key components of an exponential expression.

Exponents make it easier to write expressions that would otherwise be cumbersome to write. See how much easier it is to write 310 instead of the expression on the right? 

<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mn>10</mn></msup><mo>=</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn><mo>&#x2022;</mo><mn>3</mn></math>

Exponents are a different form of writing numbers. With this new form come new properties that are unique to exponents. The next section goes over these properties, also known as the Laws of Exponents.

Laws of Exponents

The Laws of Exponents are a set of properties that pertain to all exponential expressions. Study the definition below.

To learn more about exponents, click on this link. It is a presentation that goes over the Laws of Exponents in more detail. To see examples that use the Laws of Exponents, click on this link

Make sure that you are comfortable working with exponential expressions. The kinds of SAT questions you’ll encounter will rely on a thorough understanding of these properties.


SAT Skill: Exponential Expressions

Example 1

What is the value of x?

<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>8</mn><mi>x</mi></msup><mo>=</mo><mn>32</mn></math>

In order to work with the Laws of Exponents, make sure the terms have the same base. Both 8 and 32 can be written as powers of 2:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>=</mo><msup><mn>2</mn><mn>3</mn></msup><mspace linebreak="newline"/><mn>32</mn><mo>=</mo><msup><mn>2</mn><mn>5</mn></msup></math>

Rewrite the original expression with these powers of 2:

<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><msup><mn>2</mn><mn>5</mn></msup></math>

With a common base, you can now solve for x:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnspacing="0px" columnalign="right center left"><mtr><mtd><mn>3</mn><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd><mfrac><mn>5</mn><mn>3</mn></mfrac></mtd></mtr></mtable></math>

Example 2

If 4x - 5y = 10, what is the value of this expression?

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mn>16</mn><mi>x</mi></msup><msup><mn>2</mn><mi>y</mi></msup></mfrac></math>

Find a common base for the numerator and denominator, then simplify the expression:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mn>2</mn><mrow><mn>4</mn><mi>x</mi></mrow></msup><msup><mn>2</mn><mrow><mn>5</mn><mi>y</mi></mrow></msup></mfrac><mo>=</mo><msup><mn>2</mn><mrow><mn>4</mn><mi>x</mi><mo>-</mo><mn>5</mn><mi>y</mi></mrow></msup></math>

Do you notice that the exponent for 2 is the same as the left side of the linear equation? Replace the exponents

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnspacing="0px" columnalign="right center left"><mtr><mtd><mn>4</mn><mi>x</mi><mo>-</mo><mn>5</mn><mi>y</mi></mtd><mtd><mo>=</mo></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd/><mtd/><mtd/></mtr><mtr><mtd><msup><mn>2</mn><mrow><mn>4</mn><mi>x</mi><mo>-</mo><mn>5</mn><mi>y</mi></mrow></msup></mtd><mtd><mo>=</mo></mtd><mtd><msup><mn>2</mn><mn>10</mn></msup></mtd></mtr></mtable></math>

Example 3

Rewrite this expression as an exponential expression with a power of 5.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mn>125</mn><mn>4</mn></mroot></math>

First, write the numerical expression as an exponential expression:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mn>125</mn><mn>4</mn></mroot><mo>=</mo><msup><mn>125</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></math>

Next, rewrite 125 as a power of 5:

<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>125</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></msup><mo>=</mo><msup><mfenced><msup><mn>5</mn><mn>3</mn></msup></mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></msup></math>

Finally, rewrite the entire expression:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mn>125</mn><mn>4</mn></mroot><mo>=</mo><msup><mn>5</mn><mfrac><mn>3</mn><mn>4</mn></mfrac></msup></math>

Compound Interest

A special case involving exponential expressions is compound interest. Expect to see a question of this type on the SAT. What is compound interest? Usually, the context is of money earning interest in a bank account.

Here is the compound interest formula

This is the formula to use when the compounding periods are non-continuous. Use this formula for continuous compounding:

To see examples of calculating compound interest, click on this link

SAT Skill: Compound Interest

Example 1

A savings account earns interest according the formula below. What is the exponential function that models this growth? How much is in the savings account after five years?

“Each year the amount of the account increases by 2.5%. The initial investment was $1750.”

Use the compound interest placeholder formula:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>r</mi><mi>n</mi></mfrac></mrow></mfenced><mrow><mi>n</mi><mi>t</mi></mrow></msup></math>

In this formula:

A = Current amount in savings

P = Initial investment

r = Interest rate

n = Number of compounding periods

t = Number of years

We know the following:

P = 1750

r = 0.025

n = 1

We can now generate the exponential function that models the growth:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>1750</mn><msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>025</mn></mrow></mfenced><mi>t</mi></msup></math>

We can use this model to find the amount in savings after 5 years:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnspacing="0px" columnalign="right center left"><mtr><mtd><mi>A</mi><mfenced><mn>5</mn></mfenced></mtd><mtd><mo>=</mo></mtd><mtd><mn>1750</mn><msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>025</mn></mrow></mfenced><mn>5</mn></msup></mtd></mtr><mtr><mtd/><mtd><mo>=</mo></mtd><mtd><mn>1979</mn><mo>.</mo><mn>96</mn></mtd></mtr></mtable></math>


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