Evaluate without using mathematical tables. sqrt(0.0625 2.56 / (0.25 0.08 * 0.5))

$$\sqrt{\frac{0.0625 \times 2.56}{0.25 \times 0.08 \times 0.5}}$$

Tunajua kuwa:
$0.0625 = \frac{625}{10000} = \frac{1}{16}$
$2.56 = \frac{256}{100}$
$0.25 = \frac{25}{100} = \frac{1}{4}$
$0.08 = \frac{8}{100}$
$0.5 = \frac{5}{10} = \frac{1}{2}$

Numerator:

$$\frac{1}{16} \times \frac{256}{100} = \frac{256}{16 \times 100} = \frac{16}{100}$$

Denominator:

$$\frac{1}{4} \times \frac{8}{100} \times \frac{1}{2} = \frac{1 \times 8 \times 1}{4 \times 100 \times 2} = \frac{8}{800} = \frac{1}{100}$$

$$\sqrt{16} = 4$$

Jibu la mwisho ni $\boxed{\text{4}}$.

5. Solve for $x$ in the equation $(3^{2x})^3 = 81 \times 9^4$.
Tunahitaji kuandika pande zote mbili za mlinganyo kwa kutumia msingi mmoja, ambao ni 3.

$$81 = 3^4$$ $$9 = 3^2$$

Hivyo, $9^4 = (3^2)^4 = 3^{2 \times 4} = 3^8$.

$$81 \times 9^4 = 3^4 \times 3^8$$

Tumia sheria ya vipeo $a^m \times a^n = a^{m+n}$:

$$3^4 \times 3^8 = 3^{4+8} = 3^{12}$$

$$3^{6x} = 3^{12}$$

Kwa kuwa besi ni sawa, tunaweza kulinganisha vipeo.

$$6x = 12$$

$$x = \frac{12}{6}$$ $$x = 2$$

Jibu la mwisho ni $\boxed{\text{2}}$.

6. After subtracting $\frac{3}{4}$ from the numerator of a certain fraction and subtracting from its denominator the fraction becomes $\frac{3}{5}$ (4mks)
Samahani, Victor. Swali hili halijakamilika. Sehemu inayosema "and subtracting from its denominator" haijaeleza ni kiasi gani kinapaswa kutolewa kutoka kwa denominator. Tafadhali thibitisha kiasi kinachotolewa kutoka kwa denominator ili niweze kukusaidia kulitatua.