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Exponents and Powers - Laws of Exponents

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

๐Ÿ”‘Concepts

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Definition of Exponents: An exponent tells us how many times a base number is multiplied by itself. Visually, this is represented as a large number (the base) with a smaller number positioned at its top-right corner (the exponent or power), such as ana^n where aa is the base and nn is the exponent.

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Multiplying Powers with the Same Base: When multiplying two expressions that have the same base, you keep the base as it is and add the exponents together. For example, if you have two terms side-by-side like 23ร—242^3 \times 2^4, you simply combine them into 23+42^{3+4}.

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Dividing Powers with the Same Base: When dividing two expressions with the same base, you keep the base and subtract the exponent of the denominator from the exponent of the numerator. This can be visualized as a fraction where common factors in the numerator and denominator cancel each other out, leaving the difference in the exponents.

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Power of a Power: If a base already raised to a power is then raised to another power, like (am)n(a^m)^n, you multiply the exponents together. Visually, this looks like an exponential expression enclosed in parentheses with another small number sitting outside at the top-right.

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Multiplying Powers with the Same Exponents: If different bases are raised to the same exponent and multiplied, you can multiply the bases first and then apply the common exponent to the result. This looks like (aร—b)m(a \times b)^m where the power is shared by the product inside the brackets.

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Dividing Powers with the Same Exponents: If two different bases are being divided but share the same exponent, the division can be performed on the bases first, and the exponent is applied to the whole quotient. This is often written as a fraction inside parentheses with a single exponent outside.

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The Zero Exponent Law: Any non-zero number raised to the power of 00 is always equal to 11. No matter how large the base looks, if the tiny number at the top-right is 00, the entire value collapses to 11.

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Numbers with Negative Bases: If a negative number is raised to an even power, the result is positive. If a negative number is raised to an odd power, the result is negative. For example, (โˆ’1)2=1(-1)^2 = 1 while (โˆ’1)3=โˆ’1(-1)^3 = -1.

๐Ÿ“Formulae

amร—an=am+na^m \times a^n = a^{m+n}

amรทan=amโˆ’na^m \div a^n = a^{m-n} (where m>nm > n)

(am)n=amร—n(a^m)^n = a^{m \times n}

amร—bm=(aร—b)ma^m \times b^m = (a \times b)^m

ambm=(ab)m\frac{a^m}{b^m} = (\frac{a}{b})^m

a0=1a^0 = 1 (where aโ‰ 0a \neq 0)

(โˆ’1)n=1(-1)^n = 1 if nn is even, and (โˆ’1)n=โˆ’1(-1)^n = -1 if nn is odd

๐Ÿ’กExamples

Problem 1:

Simplify and write the answer in exponential form: (23)4ร—25(2^3)^4 \times 2^5

Solution:

Step 1: Apply the 'Power of a Power' law to (23)4(2^3)^4. (23)4=23ร—4=212(2^3)^4 = 2^{3 \times 4} = 2^{12}. Step 2: Now multiply this result by 252^5 using the 'Multiplying Powers with Same Base' law. 212ร—25=212+5=2172^{12} \times 2^5 = 2^{12+5} = 2^{17}.

Explanation:

We first simplified the term with nested exponents by multiplying them, then combined the remaining terms by adding the exponents because the bases were the same.

Problem 2:

Simplify the expression: 25ร—52ร—t8103ร—t4\frac{25 \times 5^2 \times t^8}{10^3 \times t^4}

Solution:

Step 1: Express 2525 and 1010 in terms of their prime factors. 25=5225 = 5^2 and 103=(2ร—5)3=23ร—5310^3 = (2 \times 5)^3 = 2^3 \times 5^3. Step 2: Substitute these back into the expression: 52ร—52ร—t823ร—53ร—t4\frac{5^2 \times 5^2 \times t^8}{2^3 \times 5^3 \times t^4} Step 3: Combine the powers of 55 in the numerator: 52+2ร—t823ร—53ร—t4=54ร—t823ร—53ร—t4\frac{5^{2+2} \times t^8}{2^3 \times 5^3 \times t^4} = \frac{5^4 \times t^8}{2^3 \times 5^3 \times t^4} Step 4: Use the division law (amรทan=amโˆ’na^m \div a^n = a^{m-n}) for base 55 and base tt: 54โˆ’3ร—t8โˆ’423=51ร—t423=5t48\frac{5^{4-3} \times t^{8-4}}{2^3} = \frac{5^1 \times t^4}{2^3} = \frac{5t^4}{8}

Explanation:

The approach involves breaking down composite numbers into prime bases, applying the product law for the numerator, and finally using the division law to subtract exponents for like bases.