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IB0-1A: Indices

Last updated over 3 years ago
103 questions
Note from the author:
Index laws with integer indices.
#1
Simplify the following using the index laws.
1

3^2 \times 3^5

1

x^6 \times x^3

1

x^5 \times x^n

1

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

1

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

1

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

1

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

1

t^{3m} \div t

1

\left(5^3\right)^2

1

\left(t^4\right)^3

1

\left(y^3\right)^m

1

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

#2
Express in simplest form with a prime number base.
1

121

1

32

1

81

1

4^2

1

25^2

1

7^t \times 49

1

3^a \div 9

1

8^p \div 4

1

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

1

\dfrac{9}{3^x}

1

\left(25^t\right)^2

1

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

1

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

1

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

1

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

1

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

#3
Remove the brackets.
1

\left(xy\right)^2

1

\left(ab\right)^3

1

\left(xyz\right)^2

1

\left(3b\right)^3

1

\left(5a\right)^4

1

\left(10xy\right)^5

1

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

1

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

1

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

1

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

1

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

#4
Simplify the following expressions using one or more of the index laws.
1

4b^2 \times 2b^3

1

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

1

3ab^2 \times 2a^3

1

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

1

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

1

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

1

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

1

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

#5
Simplify, given your answers in simplest rational form.
1

3^0

1

6^{-1}

1

4^{-1}

1

5^0

1

4^2

1

4^{-2}

1

5^3

1

5^{-3}

1

7^2

1

7^{-2}

1

10^3

1

10^{-3}

#6
Simplify, giving your answers in simplest rational form.
1

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

1

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

1

2t^0, for t \neq 0.

1

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

1

7^0

1

3 \times 4^0

1

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

1

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

1

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

1

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

1

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

1

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

1

2^{0}+2^{1}

1

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

1

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

1

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

1

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

1

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

1

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

1

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

#7
Write the following without brackets or negative indices.
1

(3b)^{-1}

1

3b^{-1}

1

7a^{-1}

1

(7a)^{-1}

1

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

1

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

1

(5t)^{-2}

1

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

1

xy^{-1}

1

(xy)^{-1}

1

xy^{-3}

1

(xy)^{-3}

1

(3pq)^{-1}

1

3(pq)^{-1}

1

3pq^{-1}

1

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

1

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

1

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

1

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

1

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

#8
Use the index laws to prove that, for positive real numbers a and b, and integer n, the following equations are true.
1

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

1

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

#9
1

Find the smaller of 2^{125} and 3^{75} without a calculator.
Hint: 2^{125} = (2^5)^{25}.

#10
1

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.

  1. 2^{90}
  2. 5^{36}
  3. 3^{60}
  4. 10^24