Class 9 - Exponents for Real Numbers | Rules, Solved Examples and Practice Problems

Laws of exponents for real numbers are rules according to which we can solve any arithmetic expression that include exponents. These rules make it easy for us to perform any arithmetic operation on exponents including addition, subtraction, multiplication and division. The exponent rules are highly useful to simplify expressions with fraction exponents or decimal numbers. Let’s explore these rules of exponents based on different kinds of numbers.

Table of Contents

• What are Exponent Rules for Real Numbers
• Solved Examples on Exponent Rules of Real Numbers
• Sample Problems on Exponents Rules of Real Numbers
  
  
• Frequently Asked Questions on Exponents Rules of Real Number

What are Exponent Rules for Real Numbers

Exponent rules of real numbers is a set of exponent laws that are applied for solving expressions with exponents of decimal, fractions, irrational numbers and negative numbers. Here are the laws of exponents explained in detail:

Product Rule of Exponents: The product rule of exponents is applied when we multiply the expressions with the same base. As per this law the when two expressions with same base are multiplied with each other their exponents are added. That is, if $a^n$ and $a^m$ are two exponents with same base then,

$a^n \times a^m = a^{{n+m}}$

Quotient Rule of Exponents: The quotient rule of exponents is applied when we divide the expressions with the same base. As per this law the when two expressions with same base are divided with each other their exponents are subtracted. That is, if $x^n$ and $x^m$ are two exponents with same base then,

$\frac{{a^m}}{{a^n}} = a^{{m-n}}$

Power of Power Rule: The power of power rule of exponents is applied when the exponent of the base has another exponent i.e., $(a^m)^n$ then these exponents are multiplied to each other.

$(a^m)^n = a^{{m \times n}}$

Power of Product Rule: The power of product rule of exponents is applied when product is raised to a power. As per this rule, the power is distributed to each multiplicand of the product.

$(ab)^n = a^n b^n$

Power of Quotient Rule: Power of Quotient Rule is applied to the result of a quotient raised to a power. As per this law when the numerator and denominator have the same exponent it is distributed over both. For example, $\left(\frac{{a}}{{b}}\right)^n = \frac{{a^n}}{{b^n}}$

Zero Exponent Rule: Zero exponent rule states that when a base has exponent zero then it is equal to one. i.e., $a^0 = 1$

Negative Exponent Rule: As per the negative exponent rule when a base is raised to a negative exponent, it is written as its reciprocal to make the exponent positive. i.e., $b^{{-n}} = \frac{{1}}{{b^n}}$

Read more:

• Class 9 - Operations on Real Numbers
• Class 9 - Decimal Expansion of Real Numbers

Solved Examples on Exponent Rules for Real Numbers

Example 1: Find the product of $17^2$ and $17^5$ as per the laws of exponents.

Solution: As per the product law of exponents when two expressions with same base are multiplied with each other their exponents are added:

$17^2 \cdot 17^5 = 17^{{2+5}} = 17^7$

Example 2: Evaluate the value of $(5^2)^7$.

Solution: When the exponent of the base has another exponent then these exponents are multiplied by each other. So, $(5^2)^7 = 5^{{2 \times 7}} = 5^{{14}}$

Example 3: What is the value of $\frac{{23^{{10}}}}{{23^6}}$?

Solution: $\frac{{23^{{10}}}}{{23^6}} = 23^{{10-6}} = 23^2$

Sample Problems on Exponents Rules for Real Numbers

Evaluate the value of following:

1. $(6^{{-1}} + 12^{{-1}}) \times 5^{{-1}}$
2. $(2^{{-2}} \times 3^{{-2}}) + 7^{{-1}}$