Laws of Indices (Same Base, Same Indices)

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(0:01) What are Laws of Indices?: Laws of Indices are rules that simplify calculations involving powers of numbers.
(0:21) Multiplying Indices with Same Base: For multiplication, add the indices ( $a^{m} \times a^{n} = a^{m + n}$ ). For example, $2^{3} \times 2^{4} = 2^{3 + 4} = 2^{7}$ .
(1:10) Dividing Indices with Same Base: For division, subtract the indices ( $a^{m} \div a^{n} = a^{m - n}$ ). For example, $4^{5} \div 4^{3} = 4^{5 - 3} = 4^{2}$ .
(2:02) Multiplying Indices with Same Index: For multiplication, multiply the bases and keep the index ( $a^{m} \times b^{m} = (a \times b)^{m}$ ). For example, $2^{3} \times 5^{3} = (2 \times 5)^{3} = 10^{3}$ .
(2:56) Dividing Indices with Same Index: For division, divide the bases and keep the index ( $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ ). For example, $\frac{6^{5}}{3^{5}} = {(\frac{6}{3})}^{5} = 2^{5}$ .

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Laws of indices chart showing rules for multiplying and dividing powers with the same base and same index, with examples for both operations. — Law of Indices Summary

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Laws of Indices (Same Base or Same Indices)

1 / 6

Q: What is $2^{3} \times 2^{4}$ ?

2^{6}

2^{7}

2^{12}

2^{8}

Feedback: Not quite! When multiplying numbers with the same base, we add the exponents. The formula used is $a^{m} \times a^{n} = a^{m + n}$ . So, $2^{3} \times 2^{4} = 2^{3 + 4} = 2^{7}$ .

Feedback: Well done! When multiplying numbers with the same base, we add the exponents. The formula used is $a^{m} \times a^{n} = a^{m + n}$ . So, $2^{3} \times 2^{4} = 2^{3 + 4} = 2^{7}$ .

2 / 6

Q: What is $4^{5} \div 4^{3}$ ?

4^{15}

4^{8}

4^{3}

4^{2}

Feedback: Not quite! When dividing numbers with the same base, subtract the exponents. The formula is $\frac{a^{m}}{a^{n}} = a^{m - n}$ . So, $4^{5} \div 4^{3} = 4^{5 - 3} = 4^{2}$ .

Feedback: Well done! When dividing numbers with the same base, subtract the exponents. The formula is $\frac{a^{m}}{a^{n}} = a^{m - n}$ . So, $4^{5} \div 4^{3} = 4^{5 - 3} = 4^{2}$ .

3 / 6

Q: What is $10^{5} \div 10^{2}$ ?

10^{1}

10^{3}

10^{2.5}

10^{2}

Feedback: Not quite! When dividing numbers with the same base, subtract the exponents. The formula is $\frac{a^{m}}{a^{n}} = a^{m - n}$ . So, $10^{5} \div 10^{2} = 10^{5 - 2} = 10^{3}$ .

Feedback: Well done! When dividing numbers with the same base, subtract the exponents. The formula is $\frac{a^{m}}{a^{n}} = a^{m - n}$ . So, $10^{5} \div 10^{2} = 10^{5 - 2} = 10^{3}$ .

4 / 6

Q: Simplify ${0.2}^{5} \times 5^{5}$ .

1

{0.1}^{5}

(0.2 \times 5)^{5}

5

Feedback: Not quite! Using the multiplication rule for numbers with the same index $a^{n} \times b^{n} = (a \times b)^{n}$ , we get ${0.2}^{5} \times 5^{5} = (0.2 \times 5)^{5} = 1^{5} = 1$ .

Feedback: Well done! Using the multiplication rule for numbers with the same index $a^{n} \times b^{n} = (a \times b)^{n}$ , we get ${0.2}^{5} \times 5^{5} = (0.2 \times 5)^{5} = 1^{5} = 1$ .

5 / 6

Q: Simplify $5^{4} \div 5^{6}$ .

5^{2}

5^{- 2}

5^{0}

5^{- 3}

Feedback: Not quite! When dividing numbers with the same base, subtract the exponents. The formula is $\frac{a^{m}}{a^{n}} = a^{m - n}$ . So, $5^{4} \div 5^{6} = 5^{4 - 6} = 5^{- 2}$ .

Feedback: Well done! When dividing numbers with the same base, subtract the exponents. The formula is $\frac{a^{m}}{a^{n}} = a^{m - n}$ . So, $5^{4} \div 5^{6} = 5^{4 - 6} = 5^{- 2}$ .

6 / 6

Q: What is $2^{4} \times 2^{2} \times 2^{3}$ ?

2^{7}

2^{9}

2^{10}

2^{8}

Feedback: Not quite! When multiplying numbers with the same base, add the exponents. The formula is $a^{m} \times a^{n} \times a^{p} = a^{m + n + p}$ . So, $2^{4} \times 2^{2} \times 2^{3} = 2^{4 + 2 + 3} = 2^{9}$ .

Feedback: Well done! When multiplying numbers with the same base, add the exponents. The formula is $a^{m} \times a^{n} \times a^{p} = a^{m + n + p}$ . So, $2^{4} \times 2^{2} \times 2^{3} = 2^{4 + 2 + 3} = 2^{9}$ .

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