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Home / Mathematics / Laws of Exponents

SAT Laws of Exponents Definition, Rules & Examples | Testbook

Last Updated on Mar 21, 2025
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What are Exponents?

Exponents represent the number of times a number is multiplied by itself. For instance, multiplying 9 by itself three times can be represented as 93. In this case, the exponent is ‘3’, which signifies the number of times 9 is multiplied. 9 is the base, which is the number being multiplied. In essence, exponents or powers denote the number of times a number should be multiplied by itself. If the power is 2, it means the base number is multiplied twice with itself. Here are some examples:

  • 64 = 6×6×6×6
  • 115 = 11×11×11×11×11
  • 173 = 17 × 17 × 17

If a number ‘b’ is multiplied by itself n-times, it is represented as bn, where b is the base and n is the exponent.

Exponents obey certain rules, known as its laws. These laws aid in the simplification of expressions. Let's delve into the laws of exponents in more detail.

Rules of Exponents With Examples

As we have discussed, there are several laws or rules for exponents. The essential laws of exponents are as follows:

  • am × an = am+n
  • am / an = am-n
  • (am)n = amn
  • an / bn = (a/b)n
  • a0 = 1
  • a-m = 1/am
​Product With the Same Bases

According to this law, for any non-zero number a:

  • am × an = am+n

where m and n are real numbers.

Example 1: Simplify 44 × 42

Solution: 44 × 42 = 44+2 = 46

Example 2: Simplify (−3)-2 × (−3)-4

Solution: (−3)-2 × (−3)-4 = (-3)-2-4 = (-3)-6

Quotient with Same Bases

According to this rule:

  • am / an = am-n

Here, a is a non-zero number and m and n are integers.

Example 3: Simplify 5-3 / 5-1

Solution: 5-3 / 5-1 = 5-3+1 = 5-2 = 1/25

Power Raised to a Power

According to this law, if ‘a’ is the base, then the power raised to the power of base ‘a’ gives the product of the powers raised to the base ‘a’, such as;

  • (am)n = amn

Here, a is a non-zero number and m and n are integers.

Example 4: Express 162 as a power with base 4.

Solution: We have, 4×4 = 16 = 42. Therefore, 162 = (42)2 = 44.

Product to a Power

According to this rule, for two or more different bases, if the power is the same, then;

  • an bn = (ab)n

Here, a is a non-zero number and n is an integer.

Example 5: Simplify and write the exponential form of: 1/6 x 3-2

Solution: We can write, 1/6 = 2-2. Therefore, 2-2 x 3-2 = (2 x 3)-2 = 6-2.

Quotient to a Power

According to this rule, the fraction of two different bases with the same power is represented as;

  • an / bn = (a/b)n

Here, a and b are non-zero numbers and n is an integer.

Example 6: Simplify the expression and find the value: 252 / 52

Solution: We can write the given expression as; (25/5)2 = 52 = 25

Zero Power

According to this rule, when the power of any integer is zero, then its value is equal to 1, such as;

a0 = 1

Here, ‘a’ is any non-zero number.

Example 7: What is the value of 70 + 32 + 60 + 81 - 41?

Solution: 70 + 32 + 60 + 81 - 41 = 1 + 9 + 1 + 8 - 4 = 15

Negative Exponent Rule

According to this rule, if the exponent is negative, we can change the exponent into positive by writing the same value in the denominator and the numerator holds the value 1. The negative exponent rule is given as:

a-m = 1/am

Example 8: Find the value of 3-2.

Solution:

Here, the exponent is a negative value (i.e., -2). Thus, 3-2 can be written as 1/32, which equals 1/9.

Fractional Exponent Rule

The fractional exponent rule is used if the exponent is in the fractional form. The fractional exponent rule is given by:

 

 

Here, a is called the base, and 1/n is the exponent, which is in the fractional form. Thus, a1/n is said to be the nth root of a.

Example 9: Simplify: 811/2

Solution:

Here, the exponent is in fractional form. (i.e., ½). According to the fractional exponent rule, 811/2 can be written as √81, which equals 9.

Practice Problems on Laws of Exponents

Simplify the following expressions using the laws of exponents:

  1. (32)3
  2. 32 × 35
  3. 4-3
  4. 491/2
  5. 80 × 23

Conclusion

In conclusion, mastering the laws of exponents is essential for tackling math problems, especially in exams like the SAT, ACT, and GRE. These rules simplify complex expressions, making calculations quicker and easier. By understanding and applying these laws, you'll be better equipped to handle exponent-related questions confidently. So, practice these rules, and you’ll be ready to ace those tricky exponent problems in your next exam!

Frequently Asked Questions

The exponents, also called powers, define how many times we have to multiply the base number. For example, the number 2 has to be multiplied 3 times and is represented by 2^3.

The different Laws of exponents are: a^m ×a^n = a^(m+n), a^m /a^n = a^(m-n), (a^m)^n = a^(mn), a^n /b^n = (a/b)^n, a^0 = 1, a^-m = 1/a^m.

In the power of a power rule, we have to multiply the exponent values. For example, (2^3)^2 can be written as 2^6.

According to the zero power rule, if the exponent is zero, the result is 1, whatever the base value is. It means that anything raised to the power of 0 is 1. For example, 5^0 is 1.

In expression 2^2 .2^5 , the base values are the same, So we can add the exponents. Hence, 2^2 .2^5 = 2^(2+5) = 2^7.
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