IB0-1A: Indices

3^2 \times 3^5

x^6 \times x^3

x^5 \times x^n

t^3 \times t^4 \times t^5

\dfrac{7^9}{7^5}

\dfrac{x^7}{x^3}

\dfrac{t^6}{t^x}

t^{3m} \div t

\left(5^3\right)^2

\left(t^4\right)^3

\left(y^3\right)^m

\left(a^{3m}\right)^4

121

32

81

4^2

25^2

7^t \times 49

3^a \div 9

8^p \div 4

\dfrac{7^n}{7^{n-2}}

\dfrac{9}{3^x}

\left(25^t\right)^2

16^{k-3} \times 2^{-k}

\dfrac{4^a}{2^b}

\dfrac{8^x}{16^y}

\dfrac{125^{x+1}}{5^{x-1}}

\dfrac{27^{a+2}}{3^a \times 9^a}

\left(xy\right)^2

\left(ab\right)^3

\left(xyz\right)^2

\left(3b\right)^3

\left(5a\right)^4

\left(10xy\right)^5

\left(\dfrac{p}{q}\right)^2

\left(\dfrac{x}{3}\right)^4

\left(\dfrac{5}{z}\right)^3

\left(\dfrac{2a}{b}\right)^4

\left(\dfrac{3x}{4y}\right)^3

4b^2 \times 2b^3

\dfrac{a^6b^3}{a^4b}

3ab^2 \times 2a^3

\dfrac{5x^3y^2}{15xy}

\left(\dfrac{a^2}{5b}\right)^3

\dfrac{24t^6 r^4}{15t^6 r^2}

\dfrac{\left(4c^3 d^2\right)^2}{c^2 d}

\dfrac{10k^7}{(2k)^5}

3^0

6^{-1}

4^{-1}

5^0

4^2

4^{-2}

5^3

5^{-3}

7^2

7^{-2}

10^3

10^{-3}

\left(\dfrac{1}{2}\right)^0

\dfrac{5^4}{5^4}

2t^0, for t \neq 0.

\left(2t\right)^0, for t \neq 0.

7^0

3 \times 4^0

\dfrac{5^3}{5^5}

\dfrac{2^6}{2^{10}}

\dfrac{x^4}{x^9}

\left(\dfrac{3}{8}\right)^{-1}

\left(\dfrac{2}{3}\right)^{-1}

\left(\dfrac{1}{5}\right)^{-1}

2^{0}+2^{1}

5^{0} - 5^{-1}

3^{0} + 3^{1} - 3^{-1}

\left(\dfrac{1}{3}\right)^{-2}

\left(\dfrac{2}{3}\right)^{-3}

\left(1\dfrac{1}{2}\right)^{-3}

\left(\dfrac{4}{5}\right)^{-2}

\left(2\dfrac{1}{2}\right)^{-2}

(3b)^{-1}

3b^{-1}

7a^{-1}

(7a)^{-1}

\left(\dfrac{1}{t}\right)^{-2}

\left(\dfrac{3x}{y}\right)^{-1}

(5t)^{-2}

\left(5t^{-2}\right)^{-1}

xy^{-1}

(xy)^{-1}

xy^{-3}

(xy)^{-3}

(3pq)^{-1}

3(pq)^{-1}

3pq^{-1}

\dfrac{(xy)^3}{y^{-2}}

\left(5x^{-2}y^3\right)^3

\left(\dfrac{c}{2d^3}\right)^{-2}

\left(\dfrac{3r^{-3}}{t}\right)^{-2}

\left(\dfrac{2p}{5q^{-2}}\right)^{-3}

\dfrac{1}{a^{-n}}=a^n

\left(\dfrac{a}{b}\right)^{-n}=\dfrac{b^n}{a^n}

Find the smaller of 2^{125} and 3^{75} without a calculator.

Hint: 2^{125} = (2^5)^{25}.

Order the following numbers, starting with smallest at the top and largest at the bottom. Do not use a calculator. Show your work to prove how you solved this problem.