Question

Show that 7=7 using properties of exponents and radicals. Drag and drop the answer choices to the correct boxes.

Mathematical Step
(²³)² -
C
C
(37)³-1
√7=7³
=(√7) ³
= 7
Explanation
power of powers rule
product of powers rule
Inverse operations cancel each other.
transitive property
Remove the cube from both sides of the equation.
= 37.37.37
(74)³
#: (√7)³ # 7¹

Answer 1

The choices placed in the correct boxes are presented as follows;

Mathematical step Explanation

`(7^{(1)/(3) })^3` = 7¹ Power of powers rule

∛7·∛7·∛7 = (∛7)³ Product of powers rule

(∛7)³ = 7 Inverse operations cancel each other

(∛7)³ =
`\underline{(7^{(1)/(3) })^3}` Transitive property

∛7 =
`7^{(1)/(3) }` Remove the cube from both sides of the equation

The details of the reasons for the selected choices are presented as follows;

Power of powers rule states that raising a base which is raised to a power to another power is equivalent to the base raised to the product of the two powers. Mathematically, we get;

(mᵃ)ᵇ = mᵃˣᵇ

Product of powers rule states that the product of two terms with the same base is equivalent to the base being raised to sum of the exponents. Mathematically

mᵃ × mᵇ = m⁽ᵃ ⁺ ᵇ⁾

Inverse operations cancel each other; Inverse operations are mathematical operations that undo each other when applied sequentially

The inverse operation of an exponent of a value is the square root of the value

Transitive property; The transitive property states that if a = b, and a = c, then b = c