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Which two integers is $\sqrt{3^{90}}$ between? $\newline$ Choices: $\newline$ (A) $13$ and $14$ $\newline$ (B) $4$ and $5$ $\newline$ (C) $10$ and $11$ $\newline$ (D) $3$ and $4$

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Estimate cube roots

Full solution

Q. Which two integers is

\sqrt{3^{90}}

between?

\newline

Choices:

\newline

(A)

13

and

14

\newline

(B)

4

and

5

\newline

(C)

10

and

11

\newline

(D)

3

and

4

Identify Perfect Cubes: Identify the perfect cubes near $90$ . Find the perfect cubes just below and just above $90$ . The perfect cube just below $90$ is $64$ ( $4^3 = 64$ ). The perfect cube just above $90$ is $125$ ( $5^3 = 125$ ).
Calculate Cube Roots: Calculate the cube roots of $64$ and $125$ .\ Cube root of $64$ is $4$ , because $4 \times 4 \times 4 = 64$ .\ Cube root of $125$ is $5$ , because $5 \times 5 \times 5 = 125$ .
Determine Cube Root Range: Determine between which integers the cube root of $90$ falls. $\newline$ Since $64 < 90 < 125$ , it follows that $4 < \text{cube root of } 90 < 5$ .

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