Fractions and Exponents Practice: Starter
Numbers only: adding, multiplying and dividing fractions, then powers, negative exponents and roots. No calculator. If these are automatic, everything with letters gets much easier.
Work each one on paper first, then open the solution: it shows the steps, and many have a tip on the rule being used. The rules themselves are in the fractions and exponents note; the other sheets are on the resources page.
Fractions with numbers
Calculate, and give the answer as a fraction in lowest terms.
- 1.\( \displaystyle \frac{18}{24} \)
Solution
\( \displaystyle = \frac{6 \cdot 3}{6 \cdot 4} \)
\( \displaystyle = \frac{3}{4} \)Simplifying means dividing top and bottom by the same number. Here 6 divides both.
- 2.\( \displaystyle \frac{35}{84} \)
Solution
\( \displaystyle = \frac{7 \cdot 5}{7 \cdot 12} \)
\( \displaystyle = \frac{5}{12} \) - 3.\( \displaystyle \frac{2}{3} + \frac{1}{4} \)
Solution
\( \displaystyle = \frac{8}{12} + \frac{3}{12} \)
\( \displaystyle = \frac{11}{12} \)To add, first write both fractions over a common denominator, then add the numerators only.
- 4.\( \displaystyle \frac{5}{6} - \frac{3}{8} \)
Solution
\( \displaystyle = \frac{20}{24} - \frac{9}{24} \)
\( \displaystyle = \frac{11}{24} \)The smallest common denominator of 6 and 8 is 24, not 48. Either works, 24 just keeps the numbers small.
- 5.\( \displaystyle \frac{7}{10} - \frac{1}{4} \)
Solution
\( \displaystyle = \frac{14}{20} - \frac{5}{20} \)
\( \displaystyle = \frac{9}{20} \) - 6.\( \displaystyle \frac{1}{2} + \frac{1}{3} + \frac{1}{6} \)
Solution
\( \displaystyle = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} \)
\( \displaystyle = \frac{6}{6} \)
\( \displaystyle = 1 \) - 7.\( \displaystyle 3 - \frac{2}{5} \)
Solution
\( \displaystyle = \frac{15}{5} - \frac{2}{5} \)
\( \displaystyle = \frac{13}{5} \)A whole number is a fraction with denominator 1: \( 3 = \frac{15}{5} \).
- 8.\( \displaystyle -\frac{2}{3} + \frac{1}{2} \)
Solution
\( \displaystyle = -\frac{4}{6} + \frac{3}{6} \)
\( \displaystyle = -\frac{1}{6} \) - 9.\( \displaystyle \frac{3}{4} \cdot \frac{8}{9} \)
Solution
\( \displaystyle = \frac{3 \cdot 8}{4 \cdot 9} \)
\( \displaystyle = \frac{24}{36} \)
\( \displaystyle = \frac{2}{3} \)To multiply, multiply tops together and bottoms together. No common denominator needed.
- 10.\( \displaystyle \frac{5}{6} \cdot \frac{9}{10} \)
Solution
\( \displaystyle = \frac{45}{60} \)
\( \displaystyle = \frac{3}{4} \) - 11.\( \displaystyle 4 \cdot \frac{3}{8} \)
Solution
\( \displaystyle = \frac{4 \cdot 3}{8} \)
\( \displaystyle = \frac{12}{8} \)
\( \displaystyle = \frac{3}{2} \)Only the numerator gets multiplied: \( 4 \cdot \frac{3}{8} = \frac{12}{8} \), not \( \frac{12}{32} \).
- 12.\( \displaystyle \frac{2}{3} \cdot \frac{9}{4} \cdot \frac{1}{6} \)
Solution
\( \displaystyle = \frac{18}{72} \)
\( \displaystyle = \frac{1}{4} \) - 13.\( \displaystyle \frac{2}{3} \div \frac{4}{5} \)
Solution
\( \displaystyle = \frac{2}{3} \cdot \frac{5}{4} \)
\( \displaystyle = \frac{10}{12} \)
\( \displaystyle = \frac{5}{6} \)Dividing by a fraction is multiplying by its reciprocal: flip the second fraction and multiply.
- 14.\( \displaystyle \frac{3/4}{6} \)
Solution
\( \displaystyle = \frac{3}{4} \cdot \frac{1}{6} \)
\( \displaystyle = \frac{3}{24} \)
\( \displaystyle = \frac{1}{8} \)Dividing by 6 is multiplying by \( \frac16 \).
- 15.\( \displaystyle \frac{6}{3/4} \)
Solution
\( \displaystyle = 6 \cdot \frac{4}{3} \)
\( \displaystyle = \frac{24}{3} \)
\( \displaystyle = 8 \)How many quarters of three fit in 6? Eight. Dividing by a number smaller than 1 makes things bigger.
- 16.\( \displaystyle \frac{1/2}{1/4} \)
Solution
\( \displaystyle = \frac{1}{2} \cdot 4 \)
\( \displaystyle = 2 \) - 17.\( \displaystyle \frac{1}{3} + \frac{1}{3} \cdot \frac{1}{2} \)
Solution
\( \displaystyle = \frac{1}{3} + \frac{1}{6} \)
\( \displaystyle = \frac{2}{6} + \frac{1}{6} \)
\( \displaystyle = \frac{1}{2} \)Multiplication before addition, as with whole numbers.
- 18.\( \displaystyle \frac{3}{5} \cdot \frac{10}{9} - \frac{1}{3} \)
Solution
\( \displaystyle = \frac{30}{45} - \frac{1}{3} \)
\( \displaystyle = \frac{2}{3} - \frac{1}{3} \)
\( \displaystyle = \frac{1}{3} \) - 19.\( \displaystyle \frac{2}{5} \text{ of } 30 \)
Solution
\( \displaystyle = \frac{2}{5} \cdot 30 \)
\( \displaystyle = \frac{60}{5} \)
\( \displaystyle = 12 \)"Of" means multiply.
- 20.\( \displaystyle \left(\frac{2}{3}\right)^2 \)
Solution
\( \displaystyle = \frac{2^2}{3^2} \)
\( \displaystyle = \frac{4}{9} \)A power of a fraction is the power of the top over the power of the bottom.
- 21.\( \displaystyle 0.25 + \frac{3}{8} \)
Solution
\( \displaystyle = \frac{1}{4} + \frac{3}{8} \)
\( \displaystyle = \frac{2}{8} + \frac{3}{8} \)
\( \displaystyle = \frac{5}{8} \)Turn decimals into fractions first: \( 0.25 = \frac14 \), \( 0.5 = \frac12 \), \( 0.2 = \frac15 \).
- 22.\( \displaystyle \frac{1}{\frac12 + \frac13} \)
Solution
\( \displaystyle = \frac{1}{\frac{5}{6}} \)
\( \displaystyle = \frac{6}{5} \)Work out the bottom first, then flip it.
- 23.\( \displaystyle \frac{\frac12 + \frac13}{\frac14} \)
Solution
\( \displaystyle = \frac{5/6}{1/4} \)
\( \displaystyle = \frac{5}{6} \cdot 4 \)
\( \displaystyle = \frac{20}{6} \)
\( \displaystyle = \frac{10}{3} \) - 24.\( \displaystyle \frac{1 - \frac13}{1 + \frac13} \)
Solution
\( \displaystyle = \frac{2/3}{4/3} \)
\( \displaystyle = \frac{2}{3} \cdot \frac{3}{4} \)
\( \displaystyle = \frac{1}{2} \)
Powers and roots with numbers
Calculate without a calculator. Use the rules first, then compute.
- 25.\( \displaystyle 2^3 \cdot 2^4 \)
Solution
\( \displaystyle = 2^{3+4} \)
\( \displaystyle = 2^7 \)
\( \displaystyle = 128 \)Same base, multiplying: add the exponents.
- 26.\( \displaystyle \frac{3^5}{3^3} \)
Solution
\( \displaystyle = 3^{5-3} \)
\( \displaystyle = 3^2 \)
\( \displaystyle = 9 \)Same base, dividing: subtract the exponents.
- 27.\( \displaystyle \left(2^3\right)^2 \)
Solution
\( \displaystyle = 2^{3 \cdot 2} \)
\( \displaystyle = 2^6 \)
\( \displaystyle = 64 \)A power of a power: multiply the exponents.
- 28.\( \displaystyle 5^0 \)
Solution
\( \displaystyle = 1 \)Anything (except 0) to the power 0 is 1, because \( \frac{5^2}{5^2} = 5^{2-2} = 5^0 \) and also equals 1.
- 29.\( \displaystyle 2^{-3} \)
Solution
\( \displaystyle = \frac{1}{2^3} \)
\( \displaystyle = \frac{1}{8} \)A negative exponent means "one over": \( a^{-n} = \frac{1}{a^n} \). It does not make the number negative.
- 30.\( \displaystyle 10^{-2} \)
Solution
\( \displaystyle = \frac{1}{10^2} \)
\( \displaystyle = \frac{1}{100} \) - 31.\( \displaystyle 2^5 \cdot 2^{-3} \)
Solution
\( \displaystyle = 2^{5-3} \)
\( \displaystyle = 2^2 \)
\( \displaystyle = 4 \) - 32.\( \displaystyle \frac{2^{10}}{2^{12}} \)
Solution
\( \displaystyle = 2^{-2} \)
\( \displaystyle = \frac{1}{4} \) - 33.\( \displaystyle \frac{\left(3^2\right)^3}{3^5} \)
Solution
\( \displaystyle = \frac{3^6}{3^5} \)
\( \displaystyle = 3^1 \)
\( \displaystyle = 3 \) - 34.\( \displaystyle \left(\tfrac{1}{2}\right)^{-2} \)
Solution
\( \displaystyle = 2^2 \)
\( \displaystyle = 4 \)A negative exponent on a fraction flips the fraction: \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} \).
- 35.\( \displaystyle \left(\tfrac{2}{3}\right)^{-1} \)
Solution
\( \displaystyle = \frac{3}{2} \) - 36.\( \displaystyle 2^3 \cdot 5^3 \)
Solution
\( \displaystyle = (2 \cdot 5)^3 \)
\( \displaystyle = 10^3 \)
\( \displaystyle = 1000 \)Different bases, same exponent: \( a^n b^n = (ab)^n \).
- 37.\( \displaystyle \frac{6^2}{2^2} \)
Solution
\( \displaystyle = \left(\frac{6}{2}\right)^2 \)
\( \displaystyle = 3^2 \)
\( \displaystyle = 9 \) - 38.\( \displaystyle 16^{1/2} \)
Solution
\( \displaystyle = \sqrt{16} \)
\( \displaystyle = 4 \)The exponent \( \frac12 \) is the square root, \( \frac13 \) the cube root, \( \frac1n \) the n-th root.
- 39.\( \displaystyle 27^{1/3} \)
Solution
\( \displaystyle = \sqrt[3]{27} \)
\( \displaystyle = 3 \) - 40.\( \displaystyle 8^{2/3} \)
Solution
\( \displaystyle = \left(8^{1/3}\right)^2 \)
\( \displaystyle = 2^2 \)
\( \displaystyle = 4 \)Read \( a^{m/n} \) as: take the n-th root, then raise to the power m. Root first keeps the numbers small.
- 41.\( \displaystyle 4^{3/2} \)
Solution
\( \displaystyle = \left(\sqrt{4}\right)^3 \)
\( \displaystyle = 2^3 \)
\( \displaystyle = 8 \) - 42.\( \displaystyle 100^{3/2} \)
Solution
\( \displaystyle = \left(\sqrt{100}\right)^3 \)
\( \displaystyle = 10^3 \)
\( \displaystyle = 1000 \) - 43.\( \displaystyle 9^{-1/2} \)
Solution
\( \displaystyle = \frac{1}{9^{1/2}} \)
\( \displaystyle = \frac{1}{3} \)Two things at once: the minus sign means "one over", the \( \frac12 \) means square root.
- 44.\( \displaystyle 32^{-2/5} \)
Solution
\( \displaystyle = \frac{1}{\left(32^{1/5}\right)^2} \)
\( \displaystyle = \frac{1}{2^2} \)
\( \displaystyle = \frac{1}{4} \) - 45.\( \displaystyle \left(\tfrac{1}{4}\right)^{1/2} \)
Solution
\( \displaystyle = \frac{\sqrt 1}{\sqrt 4} \)
\( \displaystyle = \frac{1}{2} \) - 46.\( \displaystyle \left(\tfrac{8}{27}\right)^{2/3} \)
Solution
\( \displaystyle = \left(\frac{2}{3}\right)^2 \)
\( \displaystyle = \frac{4}{9} \) - 47.\( \displaystyle 25^{1/2} \cdot 25^{-3/2} \)
Solution
\( \displaystyle = 25^{\frac12 - \frac32} \)
\( \displaystyle = 25^{-1} \)
\( \displaystyle = \frac{1}{25} \)The exponent rules work exactly the same with fractional exponents.
- 48.\( \displaystyle \sqrt{2} \cdot \sqrt{8} \)
Solution
\( \displaystyle = \sqrt{16} \)
\( \displaystyle = 4 \)\( \sqrt a \cdot \sqrt b = \sqrt{ab} \), because it is \( a^{1/2} b^{1/2} = (ab)^{1/2} \).
- 49.\( \displaystyle 2^{-1} + 2^{-2} \)
Solution
\( \displaystyle = \frac{1}{2} + \frac{1}{4} \)
\( \displaystyle = \frac{3}{4} \)No exponent rule for sums: compute each power, then add.
- 50.\( \displaystyle \left(2^{-1} + 3^{-1}\right)^{-1} \)
Solution
\( \displaystyle = \left(\frac12 + \frac13\right)^{-1} \)
\( \displaystyle = \left(\frac56\right)^{-1} \)
\( \displaystyle = \frac{6}{5} \)The exponent \( -1 \) applies to the whole bracket, so add first and flip at the end. It is not \( 2 + 3 \).