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8th Grade - Laws of Exponents

Introduction

• Have you seen any number with another number (positive or negative) on the top right? You might have already; those are called Exponents.
• are some examples of exponents, also called exponent expressions.
• Exponents are nothing but repeated multiplication.
• In ${2}^{3}$, 2 is the base, and 3 is the exponent. In the case of $3{\left(x\right)}^{2}$, 3 is called the coefficient, x is the base, and 2 is the exponent.
• ${2}^{3}$ is read as two to the third power and also as two cubed.
• ${7}^{4}$ is read as seven to the fourth power.
• Exponents can be positive or negative.

Negative Exponents

• ${3}^{-4}$ is an example of negative exponents.
• Negative exponents can be solved by finding a reciprocal (dropping that number under 1)
• For example, ${3}^{-4}$ is same as $\frac{1}{{3}^{4}}$ which is same as $\frac{1}{81}$

• If the base and exponents are the same, you only need to add/subtract the coefficient.
• can be added, and the result is $5{x}^{2}$.
• can be subtracted, and the result is $2{x}^{2}$.
• can't be added as the exponents are different.
• can't be subtracted as the bases are different.
• can be added only by evaluating $2{\left(3\right)}^{3}$ and $3{\left(2\right)}^{2}$ separately and then adding. So it is 18 + 12, which is 30.

Multiplying Exponents

• To multiply exponents with the same base, keep the base and add the exponent.
• is the same as .
• is the same as which equates to ${3}^{6}$.
• If the bases are different, simplify each expression and then perform the multiplication.

Dividing Exponents

• To divide exponents with the same base, keep the base and subtract the exponent.
• is the same as .
• is the same as which equates to ${3}^{2}$.
• If the bases are different, simplify each expression and then perform the division.

Power of Exponents

• To raise an exponent to a power, multiply the exponents.
• ${\left({2}^{3}\right)}^{4}$ is the same as ${\text{2}}^{12}$.
• ${\left({x}^{3}\right)}^{-2}$ is the same as  ${x}^{-6}$.

Blunder Area

• can't be added directly as the exponents are different.
• can't be added directly as the bases are different.
• ${x}^{{3}^{2}}$ is ${x}^{6}$ and not ${x}^{5}$