Roots & Exponents

Question 1

21 x 21

Answer 1

441

Question 2

22 x 22

Answer 2

484

Question 3

23 x 23

Answer 3

529

Question 4

24 x 24

Answer 4

576

Question 5

25 x 25

Answer 5

625

Question 6

26 x 26

Answer 6

676

Question 7

27 x 27

Answer 7

729

Question 8

28 x 28

Answer 8

784

Question 9

29 x 29

Answer 9

841

Question 10

√2 ≈ ____

Answer 10

1.4

Question 11

√3 ≈ ____

Answer 11

1.7

Question 12

√5 ≈ ____

Answer 12

2.2

Question 13

√6 ≈ ____

Answer 13

2.4

Question 14

√7 ≈ ____

Answer 14

2.6

Question 15

√8 ≈ ____

Answer 15

2.8

Question 16

Square Root Operation Tip

Answer 16

2r² = (R + r)²
→ √2r = R + r
→ r = R / (√2 - 1)

BUT be mindful of +/- signs

Question 17

Perfect Square and Perfect Cubes

Answer 17

There are many numbers that are both perfect squares and perfect cubes

Question 18

Perfect Square

Answer 18

If x, y, z, and a are integers greater than 0 and x, y, and z are consecutive integers such that x < y < z and x²y³z²y^a is a perfect square, what is the value of a?

• Depends on whether y is a perfect square
• If y = perfect square → a can be any integer > 0
• If y ≠ perfect square → a must be an odd integer

Question 19

Approximating Roots

Answer 19

• 1,000^1/5 → 3⁵ = 243, 4⁵ = 1,024 → 1,000^1/5 is between 3 and 4 and much closer to 4

Question 20

The square root of a fraction is _____ than the fraction itself.

Answer 20

Larger

e.g., √4/9 = 2/3 > 4/9

Question 21

When presented with a large number raised to the second power, if the answers all have unique units digits, then just square the units digit of the original number to determine the units digit of the new number.

Answer 21

e.g., 18,117² → units digit = 9

Question 22

If a number is a perfect square, its unit digit will be 0, 1, 4, 5, 6, or 9.

(Trick: just go through 0-9 perfect squares if you forget this method)

Answer 22

Perfect squares never end in 2, 3, 7, or 8

Question 23

Principal Square Root

Answer 23

• The non-negative square root of a number
• When the radical symbol √ is used, we only consider the non-negative root of the number. (e.g., √144 = 12, not ±12)

Question 24

Exponent: a^x = a^y

Answer 24

In most cases, x = y, but there are three exceptions:

1. a = 1
2. a = -1
3. a = 0

If a ≠ 0, a ≠ 1, a ≠ -1, and a^x = a^y, then x = y.

Question 25

Exponent: x^y = 1 (and if x and y are integers)

Answer 25

1. x = 1 → y can be any integer 2. x = -1 → y can be any even integer 3. x = any other integer → y has to be 0

Question 26

Multiple Square Roots

Answer 26

Go from left to right and count how many radicals a number is under e. g., √√3 → 3 is under 2 radicals → 3^¼ e. g., √2√2√2: * First 2 is under 1 radical: 2^½ * Second 2 is under 2 radicals: 2^¼ * Third 2 is under 3 radicals: 2^1/8 * Answer = 2^½ x 2^¼ x 2^1/8 = 2^7/8

Question 27

Removing Radicals

Answer 27

Remove the radicals with the LCD of the indices of the radicals

Question 28

Comparing Radicals and Exponents: From Radical to Exponent

Answer 28

If x, y, and m are positive, then x \> y if and only if x^m \> y^m e.g., ⁴√4 vs. ⁵√7 → Raise both expressions to the LCD of 20 → ⁴√4 = 4⁵ = 1,024 \< ⁵√7 = 7⁴ = 2,401

Question 29

Comparing Radicals and Exponents: From Exponent to Radical

Answer 29

Scale down by raising to the power of 1 over GCF e.g., 5⁵⁰ vs. 7²⁵ → GCF of 25 → (5⁵⁰)^1/25 = 5² \> (7²⁵)^1/25 = 7

Question 30

Fractional base raised to a negative exponent

Answer 30

Flip the fraction and make the exponent positive. If x≠0 and y≠0, then (x/y)^-z = (y/z)^z e.g., (3/7)^-3 = (7/3)³

Question 31

Comparison: If 0 \< x \< 1, what is the relationship among x², x, and √x?

Answer 31

x² \< x \< √x

Question 32

Comparison: 0 \< x \< 1, n \> 0 and n even

Answer 32

xⁿ \< x e.g., (1/4)² \< 1/4

Question 33

Comparison: -1 \< x \< 0, n \> 1 and n odd

Answer 33

Result is larger: xⁿ \> x e.g., (-1/4)³ \> -1/4

Question 34

Comparison: x \> 1, 0 \< n \< 1

Answer 34

Result is smaller: xⁿ \< x e.g., 4^½ \< 4

Question 35

Comparison: 0 \< x \< 1, 0 \< n \< 1

Answer 35

Result is larger: xⁿ \> x e.g., (1/4)^½ \> 1/4

Question 36

Estimating Exponents

Answer 36

Realize that 2 numbers with the same base and exponents that differ by as little as 1 can be vastly different from each other. This difference is especially pronounced when the bases and exponents are relatively large. * e.g., Is 9⁸ - 2¹² closer to 9⁷ or 9⁸ → 9⁸

Question 37

Squaring decimals with zeros

Answer 37

0.006³ 1. Calculate the non-zero value first: 6³ = 216 2. Count the number of decimals (i.e., 3) then apply the exponent to see how many decimal points the resulting number will have (i.e., 3 x 3 = 9) 3. The number of zeros before the number in #1 is #2 minus the # of digits represented by #1 → 9 - 3 = 6 4. Answer = 0.000000216