Laws of Exponents in Mathematics
In this article, we are going to study about the different laws of exponents in Mathematics. These laws are very basic and very important to study in order to solve the mathematical equations in an easy way. So, we are going to learn what an exponent is and why the exponents have laws or rules.
What are Exponents?
When multiplying a number by itself repeatedly, exponents are employed to illustrate this. For instance, 73 can be used to represent 7 x 7 x 7. The exponent in this case is "3," denoting how many times the number 7 has been multiplied. Here, the number that is being multiplied is 7, which serves as the base. When the power is 2, the base value is multiplied by itself twice.
Examples for the exponents are:
- 23 = 2 x 2 x 2
- 54 = 5 x 5 x 5 x 5
Note: A number "a" is represented as “an” when it is multiplied by itself n times, where a is the base and n is the exponent.
Laws of Exponents
There are various exponent laws in Mathematics. Many mathematical problems that involve repeated multiplication operations can be solved using all the exponentiation principles. The principles of exponents make multiplication and division processes simpler and make it easier to answer issues.
Following is the list of the seven important laws of Exponents:
- Product of powers rule
- Quotient of powers rule
- Power of a power rule
- Power of a product rule
- Power of a quotient rule
- Zero power rule
- Negative exponent rule
Now, we are going to discuss briefly about each of the rule mentioned in the above list.
Product of Powers Rule
Expressions with the same bases are multiplied according to the product of powers rule. The exponents should always be added while keeping the base constant to multiply two equations with the same base, according to this rule. Exponents with the same base are added in accordance with this rule. In this case, the rule is helpful to combine two expressions into one by multiplying them.
Xm + Xn = Xm+n
Quotient of Power Rule
Expressions with the same bases are divided according to the Quotient of Power Rule. The exponents must always or subtracted while maintaining the same base in order to divide two expressions that have the same base, according to this rule. Without actually doing the division operation, this is useful in solving an expression. The two phrases must share the same base, and that is the sole requirement.
Xm / Xn = Xm-n
Power of Power Rule
Expressions of the form (Xm)n can be made simpler using the "power of a power law of exponents." According to this rule, we should simply multiply the exponents when we have a single base and two of them. One exponent over the other is available for the two exponents. To create a single exponent, these can simply be multiplied.
(Xm)n = Xmn
Power of Product Rule
To determine the outcome of a product raised to an exponent, apply the "power of a product rule of exponents." Distribute the exponents to every multiplicand of the product, according to this law.
(XY)m = Xm Ym
Power of Quotient Rule
To determine the outcome of a quotient raised to an exponent, one uses the power of a quotient rule of exponents. The exponent must be distributed among the numerator and denominator, according to this law. In this case, the bases are different, yet both bases' exponents are the same.
(X/Y)m = Xm / Ym
Zero Power Rule
When an expression's exponent is 0, the zero-power rule of exponents is used. According to this rule, "Any integer raised to 0 is 1 (other than 0)." The form 00 is an indeterminate form since it is not specified. This will clarify for us that the value of a zero exponents is always equal to 1, regardless of the base.
X0 = 1
Negative Exponent Rule
When an exponent is a negative value, the negative exponents rule is applied. According to this rule, the reciprocal should be used to change any negative exponent into a positive exponent. When the sign of the exponent values is changed, the expression is moved from the numerator to the denominator.
X-m = 1 / Xm