Teach Me Maths

Laws of Indices

Question 11 mark

Write 2^3\times2^4 as a power of 2.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

When multiplying powers with the same base, add their indices.

Worked solution

Write 2^3\times2^4 as a power of 2.

  1. 2^3\times2^4=2^{3+4}=2^7.

Question 21 mark

Write 7^6\div7^2 as a power of 7.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

When dividing powers with the same base, subtract the indices.

Worked solution

Write 7^6\div7^2 as a power of 7.

  1. 7^6\div7^2=7^{6-2}=7^4.

Question 31 mark

Write (5^2)^3 as a power of 5.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

A power raised to another power multiplies the indices.

Worked solution

Write (5^2)^3 as a power of 5.

  1. (5^2)^3=5^{2\times3}=5^6.

Question 41 mark

Work out 9^0.

Hint

Any non-zero number raised to the power 0 equals 1.

Worked solution

Work out 9^0.

  1. 9^0=1. For example, 9^2\div9^2=1, while subtracting indices gives 9^{2-2}=9^0.

Question 51 mark

Work out 4^{-1}.

Hint

A negative index gives the reciprocal of the corresponding positive power.

Worked solution

Work out 4^{-1}.

  1. 4^{-1}=\frac1{4}=\frac14.

Question 61 mark

Work out 2^{-4}.

Hint

First find 2 to the power 4, then take its reciprocal.

Worked solution

Work out 2^{-4}.

  1. 2^4=16, so 2^{-4}=\frac1{16}.

Question 71 mark

Work out 10^{-3}.

Hint

Use the reciprocal of 10 cubed.

Worked solution

Work out 10^{-3}.

  1. 10^3=1000, so 10^{-3}=\frac1{1000}=0.001.

Question 81 mark

Find the value of a in 6^2\times6^a=6^7.

Hint

The two indices on the left must add to 7.

Worked solution

Find the value of a in 6^2\times6^a=6^7.

  1. 2+a=7, so a=5.

Question 91 mark

Work out 2^3+2^2.

Hint

The multiplication law does not apply to a sum.

Worked solution

Work out 2^3+2^2.

  1. 2^3=8 and 2^2=4. Therefore 2^3+2^2=8+4=12.
  2. Adding the indices would be valid for multiplication, not for addition.

Question 101 mark

Write 5^2\div5^4 as a power of 5.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Subtract the second index from the first, even when the result is negative.

Worked solution

Write 5^2\div5^4 as a power of 5.

  1. 5^2\div5^4=5^{2-4}=5^{-2}.

Question 112 marks

Write 3^2\times3^5\div3^3 as a power of 3.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Combine the multiplication and division rules.

Worked solution

Write 3^2\times3^5\div3^3 as a power of 3.

  1. Add the indices for multiplication and subtract for division: 2+5-3=4.
  2. The answer is 3^4.

Question 122 marks

(4^3)^2\div4^5=4^n. Find n.

Hint

Apply the outer power before the division rule.

Worked solution

(4^3)^2\div4^5=4^n. Find n.

  1. The numerator is 4^{3\times2}=4^6.
  2. Dividing gives 4^{6-5}=4^1, so n=1.

Question 131 mark

Write (2^{-2})^3 as a power of 2.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Multiply the inner index by the outer index.

Worked solution

Write (2^{-2})^3 as a power of 2.

  1. The new index is -2\times3=-6. Therefore (2^{-2})^3=2^{-6}.

Question 142 marks

Work out 10^4\times10^{-6}.

Hint

First combine the powers of 10.

Worked solution

Work out 10^4\times10^{-6}.

  1. 10^4\times10^{-6}=10^{4-6}=10^{-2}.
  2. 10^{-2}=\frac1{100}=0.01.

Question 151 mark

A learner writes 3^2\times3^3=9^5. Choose the correct explanation.

Hint

The powers have the same base. Which part stays unchanged?

Worked solution

A learner writes 3^2\times3^3=9^5. Choose the correct explanation.

  1. The answer should be 3^5: keep the base and add the indices.
  2. The answer should be 3^6: multiply both indices.
  3. The answer should be 9^6: multiply the bases and the indices.
  1. The base stays 3, and 2+3=5. The correct answer is 3^5.

Question 16Challenge3 marks

Write \dfrac{(2^{-3}\times2^7)^2}{2^5} as a power of 2.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Work inside the brackets, apply the outer power, then divide.

Worked solution

Write \dfrac{(2^{-3}\times2^7)^2}{2^5} as a power of 2.

  1. Inside the brackets, 2^{-3}\times2^7=2^4.
  2. Squaring gives (2^4)^2=2^8.
  3. Finally 2^8\div2^5=2^{8-5}=2^3.

Question 17Challenge3 marks

\dfrac{(5^a)^3}{5^4}=5^8. Find the integer a.

Hint

Write the index on the left in terms of a.

Worked solution

\dfrac{(5^a)^3}{5^4}=5^8. Find the integer a.

  1. The left side is 5^{3a-4}. Equal powers of the same base give 3a-4=8.
  2. Thus 3a=12 and a=4.

Question 18Challenge3 marks

Write 8^2\times4^3 as a power of 2.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Write 8 and 4 as powers of 2 first.

Worked solution

Write 8^2\times4^3 as a power of 2.

  1. 8^2=(2^3)^2=2^6. Also 4^3=(2^2)^3=2^6.
  2. Therefore 8^2\times4^3=2^{6+6}=2^{12}.

Question 19Challenge3 marks

9^4=3^6\times k. Work out the value of k.

Hint

Write 9 as a power of 3 first.

Worked solution

9^4=3^6\times k. Work out the value of k.

  1. 9=3^2, so 9^4=(3^2)^4=3^{8}.
  2. Then 3^8=3^6\times k, so k=3^8\div3^6=3^{2}=9.

Question 20Challenge3 marks

Write \left(2^{-3}\div2^{5}\right)^{-1} as a power of 2.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Work out the division inside the brackets first. Then multiply the indices.

Worked solution

Write \left(2^{-3}\div2^{5}\right)^{-1} as a power of 2.

  1. Inside the brackets, 2^{-3}\div2^5=2^{-3-5}=2^{-8}.
  2. Then (2^{-8})^{-1}=2^{(-8)\times(-1)}=2^8.

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