The Laws of Exponents

The Laws of Exponents

By Joella Mae Lamsin

What is exponent?​

• An EXPONENT refers to the number of times a number is multiplied by itself.

​Mathematical Definition

​

​Merriam Webster

​

• one that champions, practices, or exemplies

• the POWER

Let Us Try !

​Simplifying Exponents

Direction: Simplify the following expressions completely by removing the exponents.

Ready?

Fill in the Blank

Simplify:

$$3^2$$  

Fill in the Blank

Simplify:

$$2^5$$  

Fill in the Blank

Simplify:

$$3^2\times3$$  

Fill in the Blank

Simplify:

$$\left(-9\right)^2$$  

Law of Exponents

Product Rule

When multiplying, ADD the EXPONENTS and KEEP the SAME BASE

Ex.

• multiply the coefficients

• ​add the exponents

PRODUCT RULE with Coefficients

Multiple Choice

Simplify:

$$b^2\times b^4$$  

• $$b^8$$  

• $$b^2$$  

• $$b^6$$  

• $$2b^4$$  

Multiple Choice

Simplify:

$$4^3\times4^5$$  

• $$4^8$$  

• $$4^{15}$$  

• $$16^8$$  

• $$16^{15}$$  

Multiple Choice

Simplify:

$$2y^6\times y^2$$  

• $$3y^8$$  

• $$3y^{12}$$  

• $$2y^8$$  

• $$2y^{12}$$  

Multiple Choice

Simplify:

$$8g^3\times4g^6$$  

• $$12g^9$$  

• $$32g^{18}$$  

• $$32g^9$$  

• $$12g^3$$  

Multiple Choice

Simplify:

$$\left(2a^4b^3\right)\left(4a^5b^2\right)$$  

• $$6ab^{14}$$  

• $$6a^9b^5$$  

• $$8a^{20}b^6$$  

• $$8a^9b^5$$  

Quotient Rule

When dividing, SUBTRACT the EXPONENTS and KEEP the SAME BASE

Ex.

• divide the coefficients

• ​subract the exponents

QUOTIENT RULE with Coefficients

Multiple Select

Simplify : $$\frac{5^6}{5^4}$$  

• $$5^{10}$$  

• $$5^{-2}$$  

• $$5^2$$  

• 25

Multiple Choice

Simplify : $$\frac{6b^5}{2b^3}$$  

• $$4b^2$$  

• $$3b^2$$  

• $$4b^{-2}$$  

• $$3b^{-2}$$  

Multiple Choice

Simplify : $$\frac{4^5\cdot4^{10}}{4^6}$$  

• $$4^{21}$$  

• $$4^{44}$$  

• $$4^9$$  

• $$4^{15}$$  

Multiple Choice

Simplify : $$\frac{20g^{15}}{5g^3}$$  

• $$4g^{12}$$  

• $$4g^5$$  

• $$15g^{12}$$  

• $$25g^{18}$$  

Multiple Choice

Simplify : $$\frac{4a^{12}b^{15}}{12a^{11}b^{10}}$$  

• $$3b^5$$  

• $$3ab^5$$  

• $$\frac{ab^5}{3}$$  

• $$\frac{b^5}{3}$$  

Power Of A Power

When you have an exponent RAISE to an exponent, MULTIPLY the EXPONENTS

Multiple Choice

Simplify:

$$\left(5^3\right)^4$$  

• $$5^{12}$$  

• $$5^7$$  

• $$5^{-1}$$  

• $$5^4$$  

Multiple Choice

Simplify:

$$\left(y^2\right)^4$$  

• $$y^{24}$$  

• $$y^6$$  

• $$y^2$$  

• $$y^8$$  

Multiple Choice

Simplify:

$$\left(5^3\right)^4\times5^5$$  

• $$25^{60}$$  

• $$5^{60}$$  

• $$25^{17}$$  

• $$5^{17}$$  

Ex.

• raise the coefficient to the power

• ​multiply exponents

Expanded Power Rule

​PRODUCT of a POWER

Multiple Choice

Simplify:

$$\left(x^2y\right)^5$$  

• $$x^7y^5$$  

• $$x^{10}y^5$$  

• $$x^2y^5$$  

• $$x^5y^5$$  

Multiple Choice

Simplify:

$$\left(6x^4\right)^3$$  

• $$36x^{12}$$  

•   $$216x^{12}$$  

• $$216x^7$$   

•   $$18x^{12}$$  

Multiple Choice

Simplify:

$$\left(a^5b^9\right)^{11}$$  

• $$a^{16}b^{20}$$  

•   $$a^6b^2$$  

• $$a^{55}b^{99}$$   

•   $$a^{154}b$$  

Multiple Choice

Simplify:

$$\left(4m^{10}n^5\right)^2$$  

• $$8m^{20}n^{10}$$  

•   $$16m^{12}n^7$$  

• $$16m^{20}n^{10}$$   

•   $$6m^{12}n^7$$  

Ex.

• raise dividend and divisor to the same exponent

• ​multiply exponents

Expanded Power Rule

​QUOTIENT of a POWER

Multiple Choice

Simplify:

$$\left(\frac{a^{12}}{b^4}\right)^4$$  

• $$\frac{a^{48}}{b^{16}}$$  

•   $$a^{32}$$  

• $$\frac{a^{16}}{b^8}$$   

•   $$a^{48}b^{16}$$  

Multiple Choice

Simplify:

$$\left(\frac{x^2y^3}{z^5}\right)^5$$  

• $$x^7y^8z^{10}$$  

•   $$\frac{x^{10}y^{15}}{z^{25}}$$  

• $$\frac{x^7y^8}{z^{10}}$$   

•   $$\frac{xy^{25}}{z^{25}}$$  

Multiple Choice

Simplify:

$$\left(\frac{2x^4}{3x^5}\right)^2$$  

• $$\frac{x^6}{2x^{10}}$$  

•   $$\frac{4x^6}{9x^7}$$  

• $$\frac{4x^8}{9x^{10}}$$   

•   $$\frac{x^6}{2x^7}$$  

Zero Exponent Rule

Anything with a power/exponent of 0 equals 1

Multiple Choice

What is $$100^0\ ?$$  

• 0

• 100

• 1

Multiple Choice

Simplify $$\left(-\frac{1}{2}\right)^0\$$  

• 1

• -1

• 0

• 12

Multiple Choice

Simplify $$7x\ \left(3x^2\right)^0$$  

• 1

• 7

• 7x

• x

Multiple Choice

Simplify $$\ a^0b^{15}$$  

• 1

• $$a$$  

• $$b^{15}$$  

• 0

Multiple Choice

Simplify $$\ \left(a^5b^{15}\right)^0$$  

• $$a^5$$  

• 1

• $$b^{15}$$  

• 0

Negative Exponent Rule

Get the RECIPROCAL of the base and change the sign of exponent to positive.

Ex.

Multiple Choice

Simplify: $$k^{-6}$$  

• $$\frac{k}{6}$$  

• $$k^6$$  

• $$\frac{6}{k}$$  

• $$\frac{1}{k^6}$$  

Multiple Choice

Simplify: $$\frac{1}{a^{-5}}$$  

• $$a^{-5}$$  

• $$a^5$$  

• $$\frac{a}{5}$$  

• $$\frac{5}{a}$$  

Multiple Choice

Simplify: $$4z^4\cdot6z^{-6}$$  

• $$\frac{24}{z^2}$$  

• $$\frac{10}{z^2}$$  

• $$24z^{-24}$$  

• $$10z^{-24}$$  

Multiple Choice

Simplify: $$x^3y^{-7}$$  

• $$x^{-3}y^7$$  

• $$\frac{x^3}{y^7}$$  

• $$\frac{x^3}{y^{-7}}$$  

• $$\frac{y^3}{x^7}$$  

Multiple Choice

Simplify: $$\left(\frac{6}{7}\right)^{-4}$$  

• $$\left(\frac{6}{7}\right)^4$$  

• $$6^4$$  

• $$\left(\frac{7}{6}\right)^4$$  

• $$\left(\frac{7}{6}\right)^{-4}$$  

Thank You !