McGraw-Hill Education GRE 2019

Dividing Exponents with the Same Base: Subtract the Exponents

■ When dividing exponential terms with the same base, keep the base

and subtract the exponents: (5(5^73 )) = 5(7−3) = 5^4. To understand why you are

subtracting the exponents, write out 5^7 and 5^3. Notice that you arrive at

(5 × 5 × 5 × 5 × 5 × 5 × 5)

(5 × 5 × 5)^. When you cancel out the common factors, you are^

left with four 5s in the numerator. Thus 5^4.

Raising a Power to a Power: Multiply the Exponents

■ When raising an exponential term to a power, to simplify the term, you

should multiply the exponents: (5^4 )^3 = 5(4×3) = 5^12. Why? (5^4 )^3 = (5^4 )(5^4 )(5^4 ).

This takes you back to the first scenario in which you were multiplying

exponential terms with the same base. Recall that in such a situation, you

should add the exponents. This will yield 5(4+4+4) = 5^12.

Multiplying and Dividing Exponents with Different Bases but the

Same Exponent: Multiply or Divide the Bases

■ When multiplying exponential terms with different bases but the same

exponent, keep the exponent and multiply the bases. Thus far, you have

looked only at situations in which the base is the same. What about when the

bases are different? You can still manipulate the expression if the exponents

are the same!

54 × 3^4 = (5 × 3)^4 = 15^4

Why are you allowed to combine the bases? Again, write them out.

54 = 5 × 5 × 5 × 5

34 = 3 × 3 × 3 × 3

Notice that when you multiply these two terms, you will end up with 4

combinations of (5 × 3), giving you (5 × 3)^4.

It is also important to notice that this rule works in reverse. If a product is raised to

an exponent, then the exponent will distribute to all the factors in the product:

(2^3 × 3^5 )^4 = (2^3 )^4 × (3^5 )^4 = 2^12 × 3^20

When dividing exponential terms with different bases but the same exponent,

keep the exponent and divide the bases:^12433 = (^124 )^3 = 3^3. To understand why, once

again expand the numerator and denominator. You arrive at:

12

4 ×

12

4 ×

12

4

You thus have (^124 ) three times, giving you (^124 )^3.

Negative Exponents: Flip the Base

When raising a number to a negative exponent, to get rid of the negative exponent,

you simply flip the base:

5 −3 = (^51 )−3 = (^15 )^3 =^1533 = 1251

(^23 )−3 = (^32 )^3 =^3233 =^278

CHAPTER 11 ■ ALGEBRA 271

03-GRE-Test-2018_173-312.indd 271 12/05/17 11:54 am