Q1(i):

Find : (i) $64^{\frac{1}{2}}$

Initial Setup & Given Expression

We are tasked with evaluating the following exponential expression:

$64^{\frac{1}{2}}$

Step 1: Theoretical Foundation of Fractional Exponents

[Per the Definition of Rational Exponents], an expression of the form $a^{\frac{m}{n}}$ is mathematically equivalent to the $n$-th root of $a$ raised to the $m$-th power, expressed as $\sqrt[n]{a^m}$.

In our specific case, the exponent is $\frac{1}{2}$. This indicates the principal square root of the base. Therefore:

$64^{\frac{1}{2}} = \sqrt[2]{64^1} = \sqrt{64}$

Step 2: Method 1 - Prime Factorization (Rigorous Approach)

To evaluate this expression algebraically without relying on memorized perfect squares, we first determine the prime factorization of the base, $64$.

• $64 \div 2 = 32$
• $32 \div 2 = 16$
• $16 \div 2 = 8$
• $8 \div 2 = 4$
• $4 \div 2 = 2$
• $2 \div 2 = 1$

Thus, the prime factorization is $64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6$.

Substituting this back into our original expression yields:

$(2^6)^{\frac{1}{2}}$

Step 3: Applying the Power of a Power Property

[Per the Laws of Exponents], specifically the Power of a Power property which states that $(a^m)^n = a^{m \cdot n}$, we multiply the inner exponent by the outer fractional exponent:

$(2^6)^{\frac{1}{2}} = 2^{6 \times \frac{1}{2}}$

$2^{6 \times \frac{1}{2}} = 2^3$

Evaluating $2^3$ gives:

$2 \times 2 \times 2 = 8$

Step 4: Method 2 - Perfect Square Identification (Direct Approach)

Alternatively, recognizing that $64$ is a perfect square provides a more direct algebraic route. We know that $8 \times 8 = 64$, meaning $64 = 8^2$.

Substituting $8^2$ into the expression:

$(8^2)^{\frac{1}{2}}$

Applying the same Power of a Power property $(a^m)^n = a^{m \cdot n}$:

$8^{2 \times \frac{1}{2}} = 8^1 = 8$

Geometric Representation of the Square Root

Geometrically, finding the square root of $64$ (i.e., $64^{\frac{1}{2}}$) is equivalent to finding the side length of a square whose total area is $64$ square units. The diagram below illustrates an $8 \times 8$ grid, visually proving that a side length of $8$ yields an area of $64$.

Final Conclusion

Through both prime factorization and perfect square identification, governed by the fundamental laws of exponents, the expression simplifies perfectly to the integer $8$.

Final Solution: 8