When a decimal is raised to a power, the number of decimal places in the result is equal to the number of places in the decimal multiplied by the exponent. For example, consider (0.12). There are two decimal places in 0.12 and 3 is the exponent. Therefore, the number of places in the power will be 3(2) = 6. The result is as follows:

The truth of this rule is evident when we recall the rule for multiplying decimals. Part of the rule states: Mark off as many decimal places in the product as there are decimal places in the factors together. If we carry out the multiplication, (0.12) x (0.12) x (0.12), it is obvious that there are six decimal places in the three factors together. The rule can be shown for any decimal raised to any power by simply carrying out the multiplication indicated by the exponent.

Finding a root of a number is the inverse of raising a number to a power. To determine the number of decimal places in the root of a perfect power, we divide the number of decimal places in the radicand by the index of the root. Notice that this is just the opposite of what was done in raising a number to a power.

Consider . The square root of 625 is 25. There are four decimal places in the radicand, 0.0625, and the index of the root is 2. Therefore, 4 + 2 = 2 is the number of decimal places in the root. We have

All of the laws of exponents may be developed directly from the definition of exponents. Separate laws are stated for the following five cases:

To illustrate the law of multiplication, we examine the following problem:

Recalling that 4³ means 4 x 4 x 4 and 4² means 4 x 4, we see that 4 is used as a factor five times. Therefore 4³ x 4² is the same as 4⁵. This result could be written as follows:

Notice that three of the five 4s came from the expression 4³, and the other two 4s came from the expression 4². Thus we may rewrite the problem as follows:

The law of exponents for multiplication may be stated as follows: To multiply two or more powers having the same base, add the exponents and raise the common base to the sum of the exponents. This law is further illustrated by the following examples:

It is important to realize that the base must be the same for each factor, in order to apply the laws of exponents. For example, 2³ x 3² is neither 2⁵ nor 3⁵. There is no way to apply the law of exponents to a problem of this kind. Another common mistake is to multiply the bases together. For example, this kind of error in the foregoing problem would imply that 2³ x 3² is equivalent to 6⁵, or 7776. The error of this may-be proved as follows:

The law of exponents for division may be developed from the .following example:

Cancellation of the five 6s in the divisor with five of the 6s in the dividend leaves only two 6s, the product of which is 6².

This result can be reached directly by noting that 6² is equivalent to 6^{(7 - 5)}. In other words, we have the following:

Therefore the law of exponents for division is as follows: To divide one power into another having the same base, subtract the exponent of the divisor from the exponent of the dividend.

Use the number resulting from this subtraction as the exponent of the base in the quotient. Use of this rule sometimes produces a negative exponent or an exponent whose value is 0. These two special types of exponents are discussed later in this chapter.

Consider the example (3²)⁴. Remembering that an exponent shows the number of times the base is to be taken as a factor and noting in this case that 3² is considered the base, we have

The laws of exponents for the power of a power may be stated as follows: To find the power of a power, multiply the exponents. It should be noted that this case is the only one in which multiplication of exponents is performed.

Consider the example (3 x 2 x 5)³. We know that

Thus 3, 2, and 5 appear three times each as factors, and we can show this with exponents as 3³, 2³, and 5³. Therefore, 

(3 x 2 x 5)³ = 3² x 2³ x 5³

The law of exponents for the power of a product is as follows: The power of a product is equal to the product obtained when each of the original factors is raised to the indicated power and the resulting powers are multiplied together.