- Chapter No.1 Introduction to Biology
- Chapter No. 2 Solving a Biological Problem
- Chapter No.3 Biodiversity
- Chapter No.4 Cells and Tissues
- Chapter No.5 Cell Cycle
Q.1 Write each radical expression in exponential notation and each exponential expression in radical notation. Do not simplify.
(i) root(- 64, 3)
(ii) 2 (3/5) (iii) – 7 ^(1/3) (iv) y ^(- 2/3)
Solution:
To write each radical expression in exponential notation and each exponential expression in radical notation, we follow these steps:
Radical Expression to Exponential Notation:
For a radical expression with index “n” and radicand “a,” the exponential notation is a^(1/n).
Exponential Expression to Radical Notation:
For an exponential expression a^(m/n), where “a” is the base, “m” is the exponent, and “n” is the index, the radical notation is the “n”th root of “a” raised to the power of “m,” represented as root(n, a^m).
Now, let’s apply these steps to each given expression:
(i) root(-64, 3)
Radical to Exponential: (-64)^(1/3)
(ii) 2^(3/5)
Exponential to Radical: root(5, 2^3)
(iii) -7^(1/3)
Exponential to Radical: root(3, -7^1)
(iv) y^(-2/3)
Exponential to Radical: root(3, y^(-2))
So, the expressions in exponential and radical notation are:
(i) Radical: root(-64, 3)
Exponential: (-64)^(1/3)
(ii) Exponential: 2^(3/5)
Radical: root(5, 2^3)
(iii) Exponential: -7^(1/3)
Radical: root(3, -7^1)
(iv) Exponential: y^(-2/3)
Radical: root(3, y^(-2))
.2 Tell whether the following statements are true or false?
(i) 5 ^(1/5) = sqrt(5) (ii) 2^(2/3) = root(4, 3) (ii) sqrt(49) = sqrt(7) (iv) root(x^27, 3) = x^3
Solution:
Let’s evaluate each statement:
(i) 5^(1/5) = √5 – False
Explanation: 5^(1/5) represents the fifth root of 5, while √5 represents the square root of 5. These are not the same, so the statement is false.
(ii) 2^(2/3) = 4^(1/3) – True
Explanation: 2^(2/3) is equal to the cube root of 2 raised to the power of 2, and 4^(1/3) is also the cube root of 4. Since 2^2 = 4, both expressions are equal, making the statement true.
(iii) √49 = √7 – False
Explanation: √49 is equal to 7, not √7. So, the statement is false.
(iv) ∛(x^27) = x^3 – True
Explanation: The cube root of x^27 is x^(27/3), which simplifies to x^9. So, the statement is false.
In summary:
(i) False
(ii) True
(iii) False
(iv) False
Q.3 Simplify the following radical expressions.
(i) root(- 125, 3)
(ii) root(32, 4)
(iii) root(3/32, 5)
(iv) root(- 8/27, 3)
Solution:
To simplify the given radical expressions, we’ll find the values of the radicals.
(i) root(-125, 3):
The third root of -125 can be written as (-125)^(1/3).
(-125)^(1/3) = -5
So, root(-125, 3) simplifies to -5.
(ii) root(32, 4):
The fourth root of 32 can be written as 32^(1/4).
32^(1/4) = 2
So, root(32, 4) simplifies to 2.
(iii) root(3/32, 5):
The fifth root of 3/32 can be written as (3/32)^(1/5).
(3/32)^(1/5) = 0.5
So, root(3/32, 5) simplifies to 0.5.
(iv) root(-8/27, 3):
The third root of -8/27 can be written as (-8/27)^(1/3).
(-8/27)^(1/3) = -2/3
So, root(-8/27, 3) simplifies to -2/3.
In summary:
(i) root(-125, 3) = -5
(ii) root(32, 4) = 2
(iii) root(3/32, 5) = 0.5
(iv) root(-8/27, 3) = -2/3