This worksheet provides essential practice for Class 7 students mastering the laws of exponents and prime factorization.

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Let’s go through each question step by step and solve them with clear explanations.

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Q.1 Simplify and write in exponential form:

(i) $2^5 \times 2^3$

When multiplying powers with the same base, add the exponents:
$$
2^5 \times 2^3 = 2^{5+3} = 2^8
$$

✔ Answer: $2^8$

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(ii) $p^3 \times p^2$

Same rule applies:
$$
p^3 \times p^2 = p^{3+2} = p^5
$$

✔ Answer: $p^5$

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(iii) $4^3 \times 4^2$

Again, same base (4), so add exponents:
$$
4^3 \times 4^2 = 4^{3+2} = 4^5
$$

✔ Answer: $4^5$

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(iv) $a^3 \times a^2 \times a^7$

Add all exponents:
$$
a^3 \times a^2 \times a^7 = a^{3+2+7} = a^{12}
$$

✔ Answer: $a^{12}$

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(v) $5^3 \times 5^7 \times 5^{12}$

Add exponents:
$$
5^3 \times 5^7 \times 5^{12} = 5^{3+7+12} = 5^{22}
$$

✔ Answer: $5^{22}$

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(vi) $(-4)^{100} \times (-4)^{20}$

Same base: $-4$, so add exponents:
$$
(-4)^{100} \times (-4)^{20} = (-4)^{100+20} = (-4)^{120}
$$

✔ Answer: $(-4)^{120}$

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Q.2 Simplify and write in exponential form:

This seems to be asking for simplification of addition expressions and then expressing them as powers — but adding numbers does not directly give exponential forms unless they are equal terms.

Let’s analyze each:

(i) $29 + 23 = 52$
→ Not a power of any small number. Cannot be written as a simple exponential form. But if we factor it:
$52 = 4 \times 13 = 2^2 \times 13$ → So, not a pure exponential form.

But perhaps the question meant something else? Let's assume it wants us to express the sum in terms of prime factors or see if it's a power.

Wait — maybe this is a typo? Or perhaps it's asking to simplify the expression (which is just addition), and then write the result in exponential form if possible.

So let’s try:

(i) $29 + 23 = 52$
→ $52 = 2^2 \times 13$ → Not a perfect power.

(ii) $108 + 104 = 212$
→ $212 = 2^2 \times 53$

(iii) $911 + 97 = 1008$
→ $1008 = 2^4 \times 3^2 \times 7$

(iv) $2015 + 2013 = 4028$
→ $4028 = 2^2 \times 19 \times 53$

(v) $713 + 710 = 1423$
→ $1423$ is a prime number (check: not divisible by 2,3,5,7,11,13,17,19,23,29,31 — yes, prime)

So none of these sums are perfect powers. Perhaps there was a misunderstanding.

But wait — maybe Q.2 is asking to express the sum using exponent rules — but that doesn't make sense because addition isn’t governed by exponent laws.

Alternatively, perhaps the problem meant multiplication instead of addition? Because otherwise, it's not meaningful.

But assuming the question is correct, and it wants the sum simplified and then expressed as a product of prime powers (like Q.3), then:

Let’s do that:

(i) $29 + 23 = 52 = 2^2 \times 13$
(ii) $108 + 104 = 212 = 2^2 \times 53$
(iii) $911 + 97 = 1008 = 2^4 \times 3^2 \times 7$
(iv) $2015 + 2013 = 4028 = 2^2 \times 19 \times 53$
(v) $713 + 710 = 1423$ → Prime

So likely, Q.2 is misphrased. It should probably say: "Simplify and write in prime factorization form" — which is what Q.3 does.

Let’s move on.

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Q.3 Express the following numbers as a product of powers of prime factors:

(i) 72

Factorize:
$$
72 = 8 \times 9 = 2^3 \times 3^2
$$

✔ Answer: $2^3 \times 3^2$

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(ii) 432

Divide:
$$
432 \div 2 = 216 \\
216 \div 2 = 108 \\
108 \div 2 = 54 \\
54 \div 2 = 27 \\
27 \div 3 = 9 \\
9 \div 3 = 3 \\
3 \div 3 = 1
$$

So:
$$
432 = 2^4 \times 3^3
$$

✔ Answer: $2^4 \times 3^3$

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(iii) 1000

$$
1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3
$$

✔ Answer: $2^3 \times 5^3$

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(iv) 16000

$$
16000 = 16 \times 1000 = 2^4 \times 10^3 = 2^4 \times (2 \times 5)^3 = 2^4 \times 2^3 \times 5^3 = 2^{7} \times 5^3
$$

✔ Answer: $2^7 \times 5^3$

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Q.4 Express each of the following numbers using exponential notation:

This means writing the number as a power of some base, like $2^n$, $10^n$, etc.

(i) 512

Try powers of 2:
$$
2^9 = 512
$$

✔ Answer: $2^9$

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(ii) 343

Try powers of 7:
$$
7^3 = 343
$$

✔ Answer: $7^3$

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(iii) 729

Try powers of 3:
$$
3^6 = 729
$$

✔ Answer: $3^6$

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(iv) 3125

Try powers of 5:
$$
5^5 = 3125
$$

✔ Answer: $5^5$

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Q.5 Express the following numbers as a product of powers of prime factors:

This is identical to Q.3.

So answers are:

(i) 72 = $2^3 \times 3^2$
(ii) 432 = $2^4 \times 3^3$
(iii) 1000 = $2^3 \times 5^3$
(iv) 16000 = $2^7 \times 5^3$

✔ Same as Q.3.

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Q.6 Express each of the following numbers using exponential notation:

Same as Q.4.

So:

(i) 512 = $2^9$
(ii) 343 = $7^3$
(iii) 729 = $3^6$
(iv) 3125 = $5^5$

✔ Same as Q.4.

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✔ Final Answers Summary:

#### Q.1
(i) $2^8$
(ii) $p^5$
(iii) $4^5$
(iv) $a^{12}$
(v) $5^{22}$
(vi) $(-4)^{120}$

#### Q.2
(If interpreted as sums, then only prime factorizations make sense):
(i) $52 = 2^2 \times 13$
(ii) $212 = 2^2 \times 53$
(iii) $1008 = 2^4 \times 3^2 \times 7$
(iv) $4028 = 2^2 \times 19 \times 53$
(v) $1423$ → Prime

> ⚠️ Note: This may not be what was intended. Likely, Q.2 was meant to be multiplication (e.g., $2^9 \times 2^3$), but as written, it's addition.

#### Q.3 & Q.5
(i) $72 = 2^3 \times 3^2$
(ii) $432 = 2^4 \times 3^3$
(iii) $1000 = 2^3 \times 5^3$
(iv) $16000 = 2^7 \times 5^3$

#### Q.4 & Q.6
(i) $512 = 2^9$
(ii) $343 = 7^3$
(iii) $729 = 3^6$
(iv) $3125 = 5^5$

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