GCSE Fractional indices Practice Questions

Free GCSE Fractional indices practice questions with full step-by-step worked solutions. Covers unit fractional index, square roots, cube roots, fourth roots. Practise exam-style problems and check your method.

unit fractional indexsquare rootscube rootsfourth rootsroots as powersnon-unit fractional index
GCSE Higher70 questionsStep-by-step solutions
Question 1
1 markeasy
Work out the value of 251225^{\frac{1}{2}}.
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Worked solution

  1. Recall the unit fraction index rule

    a12=aa^{\frac{1}{2}} = \sqrt{a}

    An index of one half means the square root. This is the key fact for the whole topic.

  2. Apply the rule

    2512=2525^{\frac{1}{2}} = \sqrt{25}

    So we need the square root of 25 — the number that multiplies by itself to give 25.

  3. Evaluate the root

    25=5\sqrt{25} = 5

    5 times 5 is 25, so the answer is 5.

Answer
55
Question 2
2 markseasy
Which is greater: 271327^{\frac{1}{3}} or 161216^{\frac{1}{2}}? You must show your working.
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Worked solution

  1. Evaluate the first power

    2713=273=327^{\frac{1}{3}} = \sqrt[3]{27} = 3

    An index of one third means the cube root, and the cube root of 27 is 3.

  2. Evaluate the second power

    1612=16=416^{\frac{1}{2}} = \sqrt{16} = 4

    An index of one half means the square root, and the square root of 16 is 4.

  3. Compare the two values

    4>3 ⇒ 1612>27134 > 3 \ \Rightarrow\ 16^{\frac{1}{2}} > 27^{\frac{1}{3}}

    4 is greater than 3, so 16 to the power one half is the greater number. On a number line 4 sits to the right of 3.

Answer
1612 is greater, because 4>316^{\frac{1}{2}} \text{ is greater, because } 4 > 3
Question 3
2 marksintermediate
Work out the value of 322532^{\frac{2}{5}}.
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Worked solution

  1. Recall the non-unit fraction rule

    amn=(an)ma^{\frac{m}{n}} = (\sqrt[n]{a})^m

    The 5 on the bottom means fifth root, and the 2 on top means square the result.

  2. Plan the two operations

    3225=(325)232^{\frac{2}{5}} = (\sqrt[5]{32})^2

    Fifth root first, then square.

  3. Take the fifth root

    25=32 ⇒ 325=22^5 = 32 \ \Rightarrow\ \sqrt[5]{32} = 2

    Five 2s multiply to 32, so the fifth root of 32 is 2.

  4. Square the result

    22=42^2 = 4

    2 squared is 4.

  5. Sense check with base 2

    3225=(25)25=2232^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^2

    Writing 32 as 2 to the power 5 and multiplying the indices gives 2 squared directly — the same answer.

  6. State the answer

    3225=432^{\frac{2}{5}} = 4

    The value is 4.

Answer
44
Question 4
3 markshard
Work out the value of 412+4−12+404^{\frac{1}{2}} + 4^{-\frac{1}{2}} + 4^0. Give your answer as a fraction.
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Worked solution

  1. Evaluate the first term

    412=4=24^{\frac{1}{2}} = \sqrt{4} = 2

    An index of one half means the square root, and the square root of 4 is 2.

  2. Evaluate the second term: handle the sign

    4−12=14124^{-\frac{1}{2}} = \frac{1}{4^{\frac{1}{2}}}

    The negative index means a reciprocal.

  3. Finish the second term

    4−12=124^{-\frac{1}{2}} = \frac{1}{2}

    One over the square root of 4 is one half.

  4. Evaluate the third term

    40=14^0 = 1

    Any non-zero number to the power 0 is 1 — the zero index rule.

  5. Set up the sum

    2+12+12 + \frac{1}{2} + 1

    Add the three values together.

  6. Add the whole numbers

    2+1=32 + 1 = 3

    Group the whole numbers first.

  7. Add the fraction

    3+12=723 + \frac{1}{2} = \frac{7}{2}

    Three is six halves, plus one more half makes seven halves.

  8. Write as a single fraction

    62+12=72\frac{6}{2} + \frac{1}{2} = \frac{7}{2}

    The question asks for a fraction, so leave it as seven halves.

  9. Sense check as a decimal

    2+0.5+1=3.52 + 0.5 + 1 = 3.5

    As decimals the sum is 3.5, which matches seven halves. ✓

  10. State the answer

    72\frac{7}{2}

    The total is seven halves.

Answer
72\frac{7}{2}
Question 5
5 markschallenging
Write x×x3\sqrt{x} \times \sqrt[3]{x} as a single power of xx.
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Worked solution

  1. Understand the goal

    x×x3=x?\sqrt{x} \times \sqrt[3]{x} = x^?

    Both factors are roots of x. Writing each as a power lets the product law combine them.

  2. Write the square root as a power

    x=x12\sqrt{x} = x^{\frac{1}{2}}

    A square root is the power one half.

  3. Write the cube root as a power

    x3=x13\sqrt[3]{x} = x^{\frac{1}{3}}

    A cube root is the power one third.

  4. Rewrite the product

    x12×x13x^{\frac{1}{2}} \times x^{\frac{1}{3}}

    The expression is now a product of two powers of x.

  5. Recall the product law

    am×an=am+na^m \times a^n = a^{m+n}

    Multiplying powers of the same base means adding the indices.

  6. Set up the index addition

    12+13\frac{1}{2} + \frac{1}{3}

    Add one half and one third — a fraction addition with different denominators.

  7. Find a common denominator

    12=36,13=26\frac{1}{2} = \frac{3}{6},\quad \frac{1}{3} = \frac{2}{6}

    Sixths work for both fractions.

  8. Add the fractions

    36+26=56\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

    Three sixths plus two sixths is five sixths.

  9. Write the single power

    x×x3=x56\sqrt{x} \times \sqrt[3]{x} = x^{\frac{5}{6}}

    So the product is x to the power five sixths.

  10. Interpret the result

    x56=x56x^{\frac{5}{6}} = \sqrt[6]{x^5}

    Five sixths means the sixth root of x to the 5 — a single root, as required.

  11. Check with a value: set x=64x = 64

    64=8,643=4\sqrt{64} = 8,\quad \sqrt[3]{64} = 4

    Test with x=64x = 64: the square root is 8 and the cube root is 4.

  12. Multiply the check values

    8×4=328 \times 4 = 32

    The product is 32.

  13. Evaluate the formula at x=64x = 64

    6456=(646)5=25=3264^{\frac{5}{6}} = (\sqrt[6]{64})^5 = 2^5 = 32

    The sixth root of 64 is 2, and 2 to the 5 is 32 — the same value. ✓

  14. Watch for the common error

    12+13≠15\frac{1}{2} + \frac{1}{3} \ne \frac{1}{5}

    Do not add tops and bottoms separately — a common fractions slip. The correct sum is five sixths.

  15. State the answer

    x56x^{\frac{5}{6}}

    As a single power, the product is x to the power five sixths.

Answer
x56x^{\frac{5}{6}}

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