GCSE Index laws in algebra Practice Questions

Free GCSE Index laws in algebra practice questions with full step-by-step worked solutions. Covers multiplication law, adding indices, division law, subtracting indices. Practise exam-style problems and check your method.

multiplication lawadding indicesdivision lawsubtracting indicespower lawmultiplying indices
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
Simplify x2×x3x^2 \times x^3.
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Worked solution

  1. Recall the multiplication law

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

    When multiplying powers of the same base, keep the base and add the indices.

  2. Add the indices

    x2+3x^{2+3}

    The base x stays the same and the indices 2 and 3 are added.

  3. State the answer

    x5x^{5}

    Two plus three is five, so the answer is x to the power 5.

Answer
x5x^{5}
Question 2
1 markeasy
Which expression is equal to x5×x3x^5 \times x^3?
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Worked solution

  1. Recall the multiplication law

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

    Multiplying powers of the same base means add the indices, not multiply them.

  2. Add the indices

    x5+3x^{5+3}

    Keep the base x and add 5 and 3.

  3. Choose the matching option

    x8x^{8}

    Five plus three is eight, so the correct option is x to the power 8.

Answer
x8x^{8}
Question 3
2 marksintermediate
A student simplifies x6÷x2x^6 \div x^2 and writes x3x^3. Which statement correctly identifies the error?
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Worked solution

  1. Read what the student did

    x6÷x2=x3x^6 \div x^2 = x^3

    The student appears to have divided the indices 6 by 2 to get 3.

  2. Recall the correct law

    am÷an=am−na^m \div a^n = a^{m-n}

    Dividing powers of the same base means subtracting the indices, not dividing them.

  3. Apply the correct rule

    x6−2x^{6-2}

    Subtract the indices: 6 minus 2.

  4. Simplify

    x4x^{4}

    Six minus two is four, so the correct answer is x to the power 4.

  5. Check with numbers

    26÷22=64÷4=16=242^6 \div 2^2 = 64 \div 4 = 16 = 2^4

    Using base 2 confirms the answer is x to the power 4, not x cubed.

  6. Choose the correct statement

    subtract the indices: 6−2=4\text{subtract the indices: } 6 - 2 = 4

    The error is dividing instead of subtracting the indices.

Answer
x4x^{4}
Question 4
3 markshard
Simplify x6x9\frac{x^6}{x^9}. Give your answer with a positive index.
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Worked solution

  1. Recall the division law

    am÷an=am−na^m \div a^n = a^{m-n}

    Subtract the indices when dividing powers of the same base.

  2. Subtract the indices

    x6−9x^{6-9}

    Keep the base x and subtract 9 from 6.

  3. Work out the index

    x−3x^{-3}

    Six minus nine is negative three.

  4. Recall the negative-index law

    a−n=1ana^{-n} = \frac{1}{a^{n}}

    A negative index means take the reciprocal.

  5. Rewrite the power

    x−3=1x3x^{-3} = \frac{1}{x^{3}}

    x to the power negative 3 becomes 1 over x cubed.

  6. Note there is no coefficient

    1x3\frac{1}{x^{3}}

    The numerator is simply 1.

  7. Write the answer

    1x3\frac{1}{x^{3}}

    The expression with a positive index is 1 over x cubed.

  8. Check by multiplying back

    1x3×x9=x6\frac{1}{x^{3}} \times x^{9} = x^{6}

    Multiplying by the denominator x to the power 9 returns x to the power 6.

  9. Confirm consistency

    x6=x6x^{6} = x^{6}

    The check agrees with the original numerator.

  10. State the answer

    1x3\frac{1}{x^{3}}

    The answer with a positive index is 1 over x cubed.

Answer
1x3\frac{1}{x^{3}}
Question 5
6 markschallenging
Simplify (2x3)2×(3x2y)212x5y\frac{(2x^3)^2 \times (3x^2 y)^2}{12x^5 y}.
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Worked solution

  1. Deal with the first bracket

    (2x3)2(2x^3)^2

    Apply the power 2 to the first bracket.

  2. Square its coefficient

    22=42^2 = 4

    Two squared is four.

  3. Multiply its indices

    x3×2=x6x^{3 \times 2} = x^{6}

    Three times two is six, so the first bracket is 4x to the power 6.

  4. Deal with the second bracket

    (3x2y)2(3x^2 y)^2

    Apply the power 2 to the second bracket.

  5. Square its coefficient

    32=93^2 = 9

    Three squared is nine.

  6. Multiply its x indices

    x2×2=x4x^{2 \times 2} = x^{4}

    Two times two is four.

  7. Multiply its y indices

    y1×2=y2y^{1 \times 2} = y^{2}

    The y has index 1, so 1 times 2 is 2, giving 9x to the power 4 y squared.

  8. Multiply the two brackets

    4x6×9x4y24x^{6} \times 9x^{4}y^{2}

    Now multiply the results of the two brackets.

  9. Multiply the coefficients

    4×9=364 \times 9 = 36

    Four times nine is thirty-six.

  10. Add the x indices

    x6+4=x10x^{6+4} = x^{10}

    Six plus four is ten, so the numerator is 36x to the power 10 y squared.

  11. Rewrite the whole fraction

    36x10y212x5y\frac{36x^{10}y^{2}}{12x^5 y}

    Now divide by twelve x to the power 5 y.

  12. Divide the coefficients

    36÷12=336 \div 12 = 3

    Thirty-six divided by twelve is three.

  13. Divide the x terms

    x10−5=x5x^{10-5} = x^{5}

    Ten minus five is five.

  14. Divide the y terms

    y2−1=y1y^{2-1} = y^{1}

    The denominator y has index 1, so 2 minus 1 gives 1, which is just y.

  15. State the answer

    3x5y3x^{5}y

    The fully simplified expression is three x to the power 5 y.

Answer
3x5y3x^{5}y

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