Question

How do you simplify $ {{8}^{\tfrac{2}{3}}} $ ?

Answer 1

Hint:
The following rule of exponent $ {{a}^{\tfrac{p}{q}}} $ = $ \sqrt[q]{{{a}^{p}}} $ = $ {{\left( \sqrt[q]{a} \right)}^{p}} $ is useful. Recall that "cube root of a" means $ {{a}^{\tfrac{1}{3}}} $ , which is also written as $ \sqrt[3]{a} $ , and "square root of a" means $ {{a}^{\tfrac{1}{2}}} $ , which is also written as $ \sqrt{a} $ . Observe that 8 is equal to "2 cube", i.e. 8 = 2 × 2 × 2.

Complete Step by step Solution:
We know that $ {{a}^{\tfrac{p}{q}}} $ = $ \sqrt[q]{{{a}^{p}}} $ = $ {{\left( \sqrt[q]{a} \right)}^{p}} $ , therefore $ {{8}^{\tfrac{2}{3}}} $ = $ \sqrt[3]{{{8}^{2}}} $ = $ {{\left( \sqrt[3]{8} \right)}^{2}} $ .
Since, 8 = 2 × 2 × 2 = $ {{2}^{3}} $ , the above value is equal to:
= $ {{\left( \sqrt[3]{{{2}^{3}}} \right)}^{2}} $
Since cube-root of a cube is the number itself, we get:
= $ {{2}^{2}} $ , which can be written as:
= 2 × 2

= 4
which is the required simplification.

Note:
In general, the notation $ {{a}^{x}} $ is used to represent the value of the product a × a × a × a ... (x times). Here a is called the base (radix) and x is called the exponent / power (index). If $ {{a}^{x}} $ = b, then we say $ {{b}^{\tfrac{1}{x}}} $ = $ \sqrt[x]{b} $ = a, which is read as "x-th root of b is equal to a". The roots are also called radicals.
When representing the numbers, for radical numbers $ a\sqrt{b} $ means "a × $ \sqrt{b} $ ", whereas for fractions $ a\dfrac{b}{c} $ means "a + $ \dfrac{b}{c} $ ".