Laws of integral indices: แบบฝึกหัด

Simplify the expression \(\frac{y^8}{y^2 \cdot y^4}\).

Simplify the following expression, leaving your answer with positive exponents only:

$$ \frac{(2x^2 y^{-3})^2}{4x^5 y^{-7}} $$

If \(5^x = a\) and \(5^y = b\), express \((0.04)^{x-2y}\) in terms of \(a\) and \(b\).<\/p>

Evaluate \( \frac{2^5}{2^8} \).

Simplify the following expression completely, expressing your answer with positive integral indices only:

$$\left( \frac{16 x^3 y^{-2}}{2 x^{-1} y^4} \right)^{-2}$$

Express \( \frac{25}{m^{-3}} \) using only positive indices, assuming \( m \neq 0 \).

Simplify the expression \( \frac{(3k^2)^3}{9k^4} \) and express your answer with positive indices.

Solve the following simultaneous equations for \( x \) and \( y \):
\( 4^x \cdot 2^y = 32 \)
\( 27^x \div 3^y = 9 \)

Consider the algebraic expression \( E = \frac{(a^{-2}b^5)^2}{a^3 b^{-4}} \), where \( a \neq 0 \) and \( b \neq 0 \).

(a) Simplify the expression \( E \), leaving your answer with positive indices only.

(b) Using your result from part (a), find the numerical value of \( E \) when \( a=2 \) and \( b=1 \).

Simplify the following expression completely:

\[ \left( \frac{4^3 x^{-2} y^5}{2^5 x^4 y^{-1}} \right)^{-2} \]

Express the final answer using positive integral indices only.