CAMBRIDGE IGCSE MATHS (0580)

Question 1: Find r when \( 5^{\frac{r}{3}} = 125 \).

Question 2: Simplify the following:
(a) \( (4pq^2)^3 \)
(b) \( \left(16x^8\right)^{-\frac{1}{4}} \)

Question 3: Simplify the expression:
\( \left(a^{\frac{1}{2}} - b^{\frac{1}{2}}\right)\left(a^{\frac{1}{2}} + b^{\frac{1}{2}}\right) \)

Question 4: Show that \( \left(\frac{1}{10}\right)^2 + \left(\frac{2}{5}\right)^2 = 0.17 \)
Write down all the steps in your working.

Question 5:
(a) \( \left(\frac{3}{8}\right)^{\frac{3}{8}} \cdot \left(\frac{3}{8}\right)^{\frac{1}{8}} = p^q \)
Find the value of \( p \) and \( q \).
(b) \( 5^{-3} + 5^{-4} = K \cdot 5^{-4} \)
Find the value of \( K \).

Question 6: Without using a calculator, show that \( \left(\frac{49}{16}\right)^{-\frac{3}{2}} = \frac{64}{343} \).
Write down all the steps in your working.

Question 7: Simplify \( \left(256w^{256}\right)^{\frac{1}{4}} \).

Question 8: Write \( \left(27x^{12}\right)^{\frac{1}{3}} \) in its simplest form.

Question 9:
(a) \( 3^x = \sqrt[4]{3^5} \), find the value of \( x \).
(b) Simplify \( \left(32y^{15}\right)^{\frac{2}{5}} \).

Question 10:
(a) Simplify \( \left(64q^{-2}\right)^{\frac{1}{2}} \).
(b) \( \frac{5^7}{5^9} = p^2 \), find \( p \).

Question 11:
\( p = 4 \times 10^5, \, q = 5 \times 10^4 \)
(a) Find \( pq \), giving your answer in standard form.
(b) Find \( \frac{q}{p} \), giving your answer in standard form.

Question 12:
(a) \( \left(2^{24}\right)^{\frac{1}{2}} = p^4 \), find the value of \( p \).
(b) Simplify \( \frac{q^2 + q^2}{q^{\frac{1}{4}} \cdot q^{\frac{1}{4}}} \).

Question 13:
(a) Simplify \( \left(3125 t^{125}\right)^{\frac{1}{5}} \).
(b) Find the value of \( p \) when \( 3^p = \frac{1}{9} \).
(c) Find the value of \( w \) when \( \frac{x^{72}}{x^w} = x^8 \).

Question 14: Simplify \( 3x^2y^3 \cdot x^4y \).

Question 15:
(a) Simplify \( \frac{x^8}{x^2} \).
(b) Simplify \( \left(\frac{x^6}{27}\right)^{\frac{1}{3}} \).

Question 16: Simplify \( 6uw^{-3} \cdot 4uw^6 \).

Question 17: Solve \( 81^x = 3 \). Find the value of \( x \).

Question 18: Solve \( 5(w + 4 \times 10^3) = 6 \times 10^4 \).

Question 19:
(a) Find the value of:
(i) \( \left(\frac{1}{4}\right)^{0.5} \),
(ii) \( (-8)^{\frac{2}{3}} \).

Question 20:
(a) Simplify \( \frac{12x^{12}}{3x^3} \).
(b) Simplify \( \left(256y^{256}\right)^{\frac{1}{8}} \).

Question 21:
(a) Find \( \left(\sqrt{5}\right)^8 \)
(b) Find \( \left(\frac{1}{27}\right)^{-\frac{2}{3}} \)

Question 22:
Simplify \( \left(16p^{16}\right)^{\frac{1}{4}} \)

Question 23:
Simplify \( \left(\frac{x^{64}}{16y^{16}}\right)^{\frac{1}{4}} \)

Question 24:
Simplify \( \left(\frac{1}{2}x^{\frac{2}{3}}\right)^3 \)

Question 25:
(a) Simplify \( \left(32x^{10}\right)^{\frac{3}{5}} \)
(b) Simplify \( \left(36x^{16}\right)^{\frac{1}{2}} \)
(c) Simplify \( \frac{36y^5}{4y^2} \)

Question 26:
(a) Simplify \( \left(27x^6\right)^{\frac{1}{3}} \)
(b) Find the value of \( \left(64x^4\right)^{0.5} \times 4x^{-2} \)